Deformation Damage of Brazed and Machined Pyramidal Micro-Trusses by Arwa Faraj Tawfeeq B.Sc., M.Sc. (Production Engineering and Metallurgy) Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy Deakin University January, 2015
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Deformation Damage of Brazed and Machined
Pyramidal Micro-Trusses
by
Arwa Faraj Tawfeeq
B.Sc., M.Sc. (Production Engineering and Metallurgy)
Submitted in fulfilment of the requirements for the degree of
Doctor of Philosophy
Deakin University
January, 2015
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Abstract
Micro-truss sandwich structures have been attracting increasing attention over
the last decade because of their lightweight attributes and their potential for
multifunctional applications. Among different truss cores proposed for the manufacture
of three-dimensional micro-truss structures, pyramidal truss cores have been recognized
as an attractive candidate. They have a unique cell architecture with mechanical
properties that are promising for a wide range of applications, such as in the aerospace,
marine and automotive industries where a lightweight material with high flexural
stiffness and a high strength to weight ratio is needed. Further development in the
manufacturing of these structures has demonstrated the effectiveness of brazing for
assembling these sandwiches, which opens new opportunities for cost-effective and
high quality truss manufacturing.
This work investigates the mechanical performance of pyramidal micro-truss
sandwich structures after deformation damage. Emphasis was given to studying the
degradation in the mechanical properties of these structures tested in compression, shear
and bending. A comprehensive tensile characterization was also performed to help
understand the deformation behavior.
In the experimental section, two different grades of aluminum alloys, namely
AA5083 and AA3003, were used to fabricate electro-discharge machined (EDM)
AA5083 and brazed AA3003 structures. Limited previous studies have investigated
AA3003 micro-truss structures, and the current work is the first to study the
performance of a pyramidal micro-truss structure made of AA5083. Mechanical tests on
micro-trusses were conducted over plastic strain strains of up to 20%, and temperatures
in the range of 25 °C to 500 °C. Pre-loaded structures were reloaded repeatedly to
investigate the degradation of the strength and stiffness of the partially damaged
structures with increasing level of strain.
Analytical modelling and Finite Element (FE) simulation were performed to
inform interpretation and analysis of deformation data. Analytical models were used to
provide a reference point for the mechanical response of the structures, whereas FE
simulation was used to enable further analysis of the effect of plasticity parameters,
such as strain hardening exponent (n) and strain rate sensitivity index (m), on the
deformation behavior of the structures.
iv
Experimental compressive results showed that increasing the temperature from
25 °C to 500 °C resulted in residual strengths of 10% and 25% for AA5083 and
AA3003, respectively, and a residual stiffness of ~3% for both structures. Furthermore,
straining both micro-truss structures to 6% dropped their stiffness to a level near that of
a foam structure. Further increase in the reload strain reduced the stiffness to below that
of foam. On the other hand, the compressive strength of both structures outperformed
that of foam at all strains, even when these structures were strained to 17%. These
results suggested that the stiffness of these structures degrades at a faster rate than the
strength does. Simulated compressive loading results indicated an insignificant effect of
n and m on the degradation of truss load bearing capacity and stiffness. When subjected
to shear load at 25 °C, the shear bearing capacity of these structures degraded at a
slower rate due to the fact that half the struts are in tension (the other half are in
compression). This highlights the possibility of shear dominated forming processes at
higher strain with limited degradation in the strength.
This work showed that although existing theories address deformation of
cellular structures as being bending-dominated or stretch-dominated, it is possible to
have a cellular structure that deforms in between these two extremes. The transition
from stretch to bending dominated behavior in the current micro-truss structures is
characterized by a rapid degradation of stiffness and a slower degradation of strength.
Furthermore, the degradation rate in shear was moderated by the struts in tension, thus
allowing forming to take place without significant degradation in strength of the
structure.
v
To My Mother, and to the Memory of My Late Father
vi
Acknowledgments
I take this opportunity to record my sincere gratitude to my supervisor Professor
Matthew Barnett for the guidance and support, and also for continually and
convincingly conveying a spirit of advanture and excitement in regard to research and
scholarship. Without his guidance and presistent help this thesis would not have been
possible. Also, I am extremely grateful and indebted to my co-supervsior Dr. Goergina
Kelly for her unwavering guidance and valuable suggestions thorughout my PhD
course. Also, I would like to acknowledge the assistance of Dr. Filip Siska for sharing
simulation expertise that contributed positively in the analysis, interpretation, and
completion of the simulation part of this work.
I also place on record my sense of gratitude to the techanical staff and
colleagues of the Institute of Frontier Materials at Deakin University who, directly or
indirectly, have lent their helping hand in this venture, especially Mohan Setty (for
helping in metallurgy analysis and tensile tests), Dale Atwell (tensile tests), David Gray
(materials composition), John Vella (brazing blocks and shear test grips, and training on
CN machine), Lynton Leigh (brazing), Rob Pow (thermal analysis) Sandy Benness and
John Robin (CAD simualtion). I also would like to extend my appreciation to the
adminstrative staff, Helen Woodall, Helen Nicole-Starry, Marilyn Fisher, Chris
Rimmer, and Margaret Kumar for their assistance and help in various occations.
I wish to express my sincere thanks to Deakin University for providing me with
all the necessary facilities. My thanks also go to the Ministry of Higher Education and
Scientifc Research in Iraq for the financial supprot, which was instrumental for
finishing my PhD.
Finally, my deepest appreciation go to my mother, sisters, and brothers for their
unceasing encourgament and support. My sincere appreciation is extended to my friends
Dr. Firas Ridha, Dr. Salem Farhan, Abeda, Fulla, Ghadeer, Iman, and Muna, who never
ceased in helping until this thesis is structured.
Arwa F. Tawfeeq
J , 2015
vii
Table of Contents
Tile Page ………………………………………………………………………. i
Access to Thesis-A …………………………………………………………… ii
Declaration …………………………………………………………………… iii
Abstract ………………………………………………………………………. iv
Dedication ……………………………………………………………………... vi
Acknowledgments …………………………………………………………….. vii
Table of Contents ……………………………………………………………… viii
List of Figures …………………………………………………………………. xiii
List of Tables …………………………………………………………………. xx
Nomenclature …………………………………………………………………. xxii
CHAPTER 1: INTRODUCTION
1.1 Background …..…………………………………………………………… 1
1.2 Thesis Outline ...…………………………………………………………… 2
CHAPTER 2: LITERATURE REVIEW
2.1 Cellular Materials ………………………………………………………….. 5
2.1.1 General Principles ………………………………………………… 5
2.1.2 Description and Comparison of Different Cell Types ……………. 7
6.4.1 Basic Load-Displacement Behavior in Shear ……………………. 155
6.4.2 Basic Load-Displacement Behavior in Four-Point Bending …….. 157
6.5 Discussion …………………………………………………………………. 160
6.5.1 Analytical Prediction of Peak Loads ……………………………… 160
6.5.2 Comparison of Damage with Compression ……………………….. 167
6.5.3 Comparison with Foams …………………………………………... 172
6.6 Conclusions ………………………………………………………………… 177
CHAPTER 7: CONCLUSIONS ………………………………………….…… 179
CHAPTER 8: CONTRIBUTIONS TO KNOWLEDGE ……………………. 183
REFERENCES ………………………………………………………………… 186
APPENDICESS ………………………………………………………………… 203
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List of Figures
2.1 (a) a schematic illustrating the two predominant topologies exhibited by cellular metals, and (b) a comparison of the shear modulus measuredon stochastic closed cell aluminum alloys ……………………………… 6
2.2 Compressive strength of cellular materials ……………………………... 7
2.6 Comparison of the compressive stresses at yield as a function of relative density. ( cy is initial yield, y is yield strength, and c
is relative density) ………………………………………………………. 13
2.7 A schematic diagram illustrating the V-die bending brake method ……. 18
2.8 Scheme for the manufacture of pyramidal lattice cores by expanded metal sheet method ……………………………………………………… 18
2.9 Stacked truss-cores structure into multi-layer PCM ……………………. 19
2.10 Schematic of the modified manufacturing process for lattice truss cores from expanded perforated sheets ……………………………………….. 21
2.11 Failure modes in the face sheets; (a) face yielding, (b) intra-cell dimpling, and (c) face wrinkling ………………………………………………….. 26
2.12 Failure modes in the core; (a) core shear, and (b) local indentation …… 26
2.13 Mechanism map of failure domains with face sheets of different materials; (a) AA6006-0, and (b) AA60061-T6. ………………………... 27
2.14 Curved truss core panel wing skin with active cooling. The wing skinsare multifunctional ………………………………………………………. 34
2.15 Curved stainless steel sandwich structure with a corrugated mesh core ... 35
2.16 Casting method for truss structure manufacturing ……………………… 36
2.17 Schematic stress-strain curves of cellular materials; (a) foam, and (b) pyramidal ……………………………………………………….. 38
2.18 Hinge formation in the middle of the strut ……………………………… 39
2.19 Normalized stiffness; (a) compression, and (b) shear ………………….. 41
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3.1 Clad layer thickness; (a) AA3003-1 side clad, and (b) AA3003-2side clad ………………………………………………………………… 47
3.2 (a) pyramidal sandwich, (b) unit cell of pyramidal core, and (c) strut cross-section configuration ………………………………..…... 49
3.3 Pyramidal micro-truss structure made by EDM technique; (a) images from [63], and (b) photo of final truss obtained in the present study …………………………………………………………… 51
3.4 (a) tube furnace, and (b) brazing path ………………………………….. 52
3.5 Double-lap joint ………………………………………………………... 53
3.6 Stainless steel holder connected with thermocouples for lab joint brazing 55
3.7 Dog-bone configuration of tensile double-lap joint specimen …………. 56
3.8 Failed double-lap joints of AA3003-2 side clad brazed at 595 °C;(a) side and (b) top view ………………………………………………... 56
3.9 (a) bending die block; and (b) folded sheet …………………………….. 58
3.10 Brazing block for micro-truss …………………………………………... 58
3.11 Failed micro-truss structures brazed at 595 °C…………………………. 59
3.12 Top-view configuration of micro-truss; (a) two full nodes, and (b) four full nodes ………………………………………………………. 60
3.13 A brazed micro-truss structure ………………………………………….. 60
3.14 Optical microscopy images of AA3003-2 after brazing at 605 °C;(a) strut at 5X magnification, and (b) joint at 5X and 10X magnifications 61
3.15 Mechanical testing instruments for tensile, compression, and shear experiments; (a) 30kN Instron, and (b) MTS 385 ………………………. 63
3.17 Experimental grips for tensile test; (a) at room temperature, and (b) at elevated temperatures …………………………………………….. 66
3.18 Experimental grips for compression test; (a) load test at 25 °C, and (b) load/reload at elevated temperatures ……………………………………. 68
3.19 Experimental grips of shear test ……………………………………….... 71
3.20 Mesh configuration; (a) compression and shear, and (b) bending ……… 73
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3.21 Buckling shape of struts and face sheets that have been described by perturbation number at different loading modes; (a) compression, (b) shear, and (c) 4-point bending ……………………………………… 74
3.22 Input files for simulation at different n ………….………………........... 76
3.23 Input files at different m and global strain rate (which refers to the basestrain rate applied on the entire structure) of 2x10-4 s-1…..………..…… 77
3.24 Boundary conditions of compression test ………………………………. 78
3.25 Boundary conditions of 4-point bending test …………………………... 79
3.26 Boundary conditions of shear test ……………………………………… 79
4.1 Diagram of tensile tests methodology …………………………………... 87
4.2 Ambient engineering stress-strain profiles of AA5083 at different strain rates; (a) 2x10-4 s-1, (b) 10-3 s-1, (c) 10-2 s-1, and (d) 10-1 s-1 ………. 88
4.3 A comparison between the UTS of AA5083 from the current and previous work ……………………………………………………………………. 89
4.4 Ambient engineering stress-strain profiles of AA3003-1 side clad at different strain rates; (a) 2x10-4 s-1, (b) 10-3 s-1, (c) 10-2 s-1,and (d) 10-1 s-1 ………………………………………………………….. 90
4.5 Engineering stress-strain profiles of AA3003-2 side clad at 25 °Cand different strain rates; (a) 2x10-4 s-1, (b) 10-3 s-1, (c) 10-2 s-1,and (d) 10-1 s-1 ………………………………………………………….. 91
4.6 Comparison of ambient 0.02% offset yield strength …………………… 92
4.7 Comparison of ambient UTS …………………………………………… 92
4.8 E - AA5083 at elevated temperatures ………… 93
4.9 - -2 side clad at elevated temperatures 93
4.10 Yield strength and UTS of AA5083 at elevated temperatures and a strain rate of 2x10-4 s-1 ………………………………………………………… 94
4.11 Yield strength and UTS of AA3003-2 side clad at elevated temperatures and strain rate of 2x10-4 s-1 ……………………………………………... 95
4.12 The dependency of strain hardening exponent on; (a) strain rate at 25 °C,and (b) temperature at strain rate 2x10-4 s-1 …………………………….. 96
4.13 The dependency of strength hardening coefficient on temperature at a
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strain rate of 2x10-4 s-1 ………………………………………………….. 97
4.14 Strain rate sensitivity calculated according to rate jump (2x10-4 to 10-2 s-) –method I; and from - tests (2x10-4 to 10-2 s-1) –method II; (a) AA5083, and (b) AA3003 ………………………………. 98
4.15 A comparison of m for AA5083 and AA3003-2 side clad …………….. 99
4.16 Effect of strain rate on the elongation of AA5083 at 25 °C……………. 100
4.17 Effect of strain rate on the elongation of AA3003 at 25 °C……………. 100
4.18 Ductility of AA5083 as a function of temperature at strain rate of 2x10-4 s-1 ………………………………………………………………… 101
4.19 Ductility of AA3003-2 side clad as a function of temperature at strain rate of 2x10-4 s-1 ……………………………………………………….... 101
4.20 Comparison of ambient engineering tensile stress-strain for parent andannealed AA3003 at strain rate of 10-3 s-1; (a) AA3003-1 side clad, and (b) AA3003-2 side clad ……………………………………………. 102
4.21 Comparison of ambient engineering tensile stress-strain for parent and annealed AA3003-2 side clad at strain rate of 10-2 s-1 ………………….. 103
4.22 Effect of annealing on the mechanical properties of AA3003 at 25 °Cand a strain rate of 10-3 s-1 ………………………………………………. 103
5.1 Diagram of compressive test methodology ……………………………... 113
5.2 Compressive load-displacement of AA5083 structure compressed at 25 °C and strain rate of 10-4 s-1 …..…………………………………... 114
5.3 A magnified bedding-in section of Figure 5.2 ………………………….. 115
5.4 Compressive load/reload cycles of AA5083 structure at 25 °C and 10-2 s-1. Photographs are for specimen #1. Prediction shown forK= 675 and n= 0.3 (from Chapter 4) using equation (5.3) ……………… 119
5.5 Compressive load/reload cycles of AA5083 structure at 300 °C and 10-2 s-1. Photographs are for specimen #1. Prediction shown for K=76 and n=0.053 (from Chapter 4) using equation (5.3) ……………... 119
5.6 Compressive load/reload cycles of AA5083 structure at 500 °C and10-2 s-1. Photographs are for specimen #2. Prediction shown for K=12.43 and n=0.02 (from Chapter 4) using equation (5.3) ……………. 120
5.7 Compressive load/reload cycles of AA3003 micro-truss structure at 25 °C and 10-2 s-1. Photographs are for specimen #1. Prediction shown for K=217 and n=0.25 (from Chapter 4) using equation (5.3) ………….. 120
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5.8 Compressive load/reload cycles of AA3003 micro-truss structure at 300 °C and 10-2 s-1. Photographs are for specimen #1. Prediction shown for K=30 and n=0.032 (from Chapter 4) using equation (5.3) …………. 121
5.9 Compressive load/reload cycles of AA3003 micro-truss structure at 500 °C and 10-2 s-1. Photographs are for specimen #1. Prediction shown for K=13.5 and n=0.023 (from Chapter 4) using equation (5.3) ……….. 121
5.10 The effect of temperature on peak load carrying capacity of AA5083 and AA3003 micro-trusses ………………………………………………….. 123
5.11 Normalized compressive stiffness (Ec/Es) of AA5083 unit cell as a function of temperature; (a) at different reload strain levels, and (b)comparison of Ec/Es in the initial loading stage …….……………………. 126
5.12 Normalized compressive stiffness (Ec/Es) of AA3003 unit cell as a function of temperature at; (a) different reload strain levels, and (b) comparison ofEc/Es in the initial loading stage …………………………………………. 127
5.13 Normalized compressive load (F/FpK) of 2x2 unit cell AA5083 structure as a function of plastic displacement at strain rate of 10-2 s-1 …………… 128
5.14 Normalized compressive load (F/FpK) of 2x2 unit cell AA3003 structureas a function of plastic displacement at strain rate of 10-2 s-1 …………… 129
5.15 Sketch of the model design with the compressive response by the FE simulation ……………………………………………………………. 131
5.16 Load/reload for different n at 25 °C and 2x10-4 s-1. Images are at acompressive strain of 15.6% ……………………………………………. 133
5.17 Normalized compressive load (F/FpK) for different n at 25 °C and strainrate of 10-4 s-1 …………………………………………….……………… 134
5.18 Compressive pK/ y as function of n at a temperature of 25 °C and strainrate of 10-4 s-1 ……………………………………….…………………... 134
5.19 Compressive load/reload curves at 25 °C and 2x10-4 s-1 at different m.Simulation images correspond to a strain of 15.6% ……………………. 135
5.20 Comparison between the normalized compressive mechanical properties of aluminum pyramidal unit cell and commercially available open cell aluminum foams at 25°C, (a) Ec/Es pK y ……………………. 139
5.21 Bending dominant feature of low density foams with single unit cell. The edges bend when loaded giving low modulus structure …………… 141
5.22 Ec/Es of aluminum pyramidal unit cell and commercially available opencell aluminium foams at equivalent r of 0.019, 25 °C and 10-2 s-1 …… 142
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5.23 pK y of aluminum unit cell and commercially available open cell aluminum foams at equivalent r of 0.019, 25 °C, and 10-2 s-1 ……..….. 143
5.24 Comparison between the normalized compressive mechanical properties of aluminum pyramidal unit cell and commercially available open cell aluminum foams with different n at 25 °C and strain rate of 2x10-4 s-1, (a) Ec/Es (Young modulus), and (b) pk/ y ……………….. 144
6.1 Diagram of shear and bending tests methodology. All tests were carriedout at 25 °C. Test conditions shown for experimental shear, simulation shear and bending are taken from sections 3.4.4.1, 3.5.7, and 3.5.6, respectively………………………………………………………………. 150
6.2 Shear-displacement curve of AA5083 structure with 2x2 unit cells at 25 °C and 8x10-3 s-1; (a) load and (b) reload ………………………….. 152
6.3 Shear-displacement curve of AA3003 structure with 2x2 unit cellsat 25 °C and 8x10-3 s-1; (a) load and (b) reload ………………………….. 154
6.4 Simulation shear results of load-displacement curves of pyramidal micro-truss structure of 2x2 unit cells with r of 0.019 at 25 °C and strain rate of 8x10-3 s-1; (a) AA5083 and (b) AA3003 …………………... 156
6.5 Load-deflection curves of AA5083 structure with 3x3 unit cellsin 4-point bending at different r . L=72 mm at 25 °C and strain rateof 2x10-4 s-1 …………………………………………………………….... 158
6.6 A schematic of AA5083 3x3 unit cells micro-truss sandwich. tf is the face sheet thickness, c is the core thickness, Hc (=16.5 mm) is the distance from the middle of the upper and lower face sheets, d is the sandwich thickness, L (=72 mm) and B (=72 mm) are the span length and width with square cross-section, respectively……………………….. 165
6.7 Block diagram of calculation algorithm for theoretical cr in bending ….. 166
6.8 Comparison between simulation and theoretical cr from 4-point bending for pyramidal unit cell structure bent at 25 °C and strain rate of 2x10-4 s-1 in a load stage. Prediction shown is for K= 600 andn= 0.3 (Figures 4.12a and 4.13) using equation (6.2). Geometricalvalues are from Appendix V……………………………………………. 167
6.9 Comparison of normalized load (F/FpK) of 2x2 unit cell pyramidal micro-truss as a function of plastic displacement in compressive and shear loading modes at 25 °C and strain rates of 10-2 s-1 and 8x10-3 s-1
respectively; (a) experimental, and (b) simulation. Images are for struts buckling in shear and compression loadings ……..……………….. 168
xviii
6.10 A comparison of FpK (simulation) and FE (estimated required to yield half of the struts) for 2x2 AA5083 structure sheared at strain rate of 8x10-3 s-1 and 25 °C. Load-displacement curve was taken fromFigure 6.4a ………………………………………………………………. 170
6.11 Sketch shows the deformation profile of strut in shear …………………. 171
6.12 Comparison between the experimental normalized mechanical properties of aluminium pyramidal unit cell and commercially available open cell aluminum foams at equivalent r at 25 °C and strain rate of 8x10-3 s-1;(a) Gc/Es and (b) pK y …………………………………………………. 174
6.13 Comparison between the simulation normalized mechanical propertiesof aluminum pyramidal unit cell and commercially available open cell aluminium foams at equivalent r of 0.019, 25°C and strain rate of 8x10-3 s-1; (a) Gc/Es and(b) pK y.………………………………….. 177
xix
List of Tables
3.1 Chemical compositions of the alloys used in the present study ………… 46
3.2 Design parameters for the present pyramidal micro-truss structures......... 50
3.3 Brazing parameters of double-lap joints ………………………………... 54
3.4 The simulation methodology of load/reload compressive tests using tensile data at 25 °C…………………………………………………….. 80
3.5 Variation of ts/tf ratios in 4-point bending ………………………………. 83
3.6 The simulation methodology and design constant parameters ofload/reload shear at 25 °C……………………………………………… 84
5.1 Theoretical prediction and experimental averaged values of strength parameters for AA5083 structure with r =0.019, As= 2.436, and I=0.287 compressed at 25 °C and 10-4 s-1 .……………………………………….. 117
5.2 Geometrical values used in the theoretical predictions ………………….. 118
5.3 Material properties and values of stress used in the theoretical prediction of load/reload compressive response at elevated temperatures and strain rate of 10-2 s-1 ……………………………………………………………. 118
5.4 Normalized compressive stiffness for load stage ……………………….. 124 5.5 Geometrical values used in the simulations ……………………………... 132
5.6 Geometrical and material properties of a single pyramidal AA6061 core… 140
5.7 Mechanical properties of a single pyramidal AA6061 core at 25 °C and strain rate of 10-3 ………………………………………………………… 141
6.1 Interpolated tensile data used in input files of shear simulation testsat strain rate of 8x10-3s-1 and room temperature ………………………… 151
6.2 Analytical prediction and experimental averaged values of FpK for 2x2 unit cells AA5083 structure with r =0.019, As =2.436, and I= 0.287 sheared at 25 °C and strain rate of 8x10-3 s-1. K and n values at strain rate of 10-2 s-1 was used in the calculation. Geometrical values arefrom Table 5.2 ……………………………….………………….……… 162
xx
6.3 Analytical prediction and experimental averaged values of FpK for 2x2 unit cells AA3003 structure with r =0.019, As=2.567, and I= 0.335 sheared at 25 °C and strain rate of 8x10-3 s-1. K and n values at strain rate of 10-2 s-1 was used in the calculation. Geometrical values arefrom Table 5.2 …………………………………………………………… 162
6.4 Analytical prediction and simulation FpK for 2x2 unit cells of pyramidal structure with r =0.019, As=1.9, and I= 0.28 sheared at 25 °C and strain rate of 8x10-3 s-1. Geometrical values are from Table 5.5 ……………..... 163
6.5 Parameters used in calculating FE for AA5083 structure.…………….…. 170
6.6 Comparison of strength parameters, geometrical and material properties of current experimental work of AA5083 and AA3003 with a singlepyramidal core with that of AA6061 sheared at 25 °C.…………………. 175
6.7 Normalized shear stiffness and strength of partially damaged single pyramidal AA6061 core at 25 °C ………………………………………... 175
xxi
Nomenclature
Ap : Plate area (mm2)
As : Cross-section area of the strut (mm2)
a : Short diagonal of the cross-section of the strut (mm)
B : Micro-truss sandwich width (mm)
b : Long diagonal of the cross-section of the strut (mm)
C : Forming parameter (MPa)
c : Core thickness (mm)
d : Sandwich thickness (mm)
Es : Young’s modulus of fully dense material (solid) (MPa)
Et : Shanley-Engesser tangent modulus (MPa)
Ec/Es : Normalized compressive stiffness (---)
FE : Estimated load required to yield half of the struts in shear (N)
FpK : Peak force of the structure in compression, shear, bending (N)
Fy : Force at yield (N)
f : is the relative volume occupied by face sheet thickness (---)
Gc : Core shear stiffness (MPa)
Gc/Es : Normalized shear stiffness (---)
Hc : Micro-truss height between the upper and lower mid points (mm)
I : Second area moment of inertia of the strut (mm4)
K : Strength coefficient (MPa)
k : Pin-joined constant (k = 2 in compression and k = 1 in bending) (---)
L : Structure length or span length (mm)
xxii
L : Unit cell length (mm)
Lc : Strut length (mm)
Leff : Effective length = Lc/2 (mm)
Leff/r : Slenderness ratio (---)
Mf : Failure moment (N.mm)
m : Strain rate sensitivity (---)
n : Strain hardening exponent (---)
r : Radius of gyration (mm)
tf : Face sheet thickness (mm)
ts : Strut thickness (mm)
Vc : Unit cell volume for a pyramidal structure (mm3)
Vs : Volume occupied by metal for solid truss (mm3)
ws : Strut width (mm)
Zp : Plastic section modulus (mm3)
Symbols
: Material density (kg/mm3)
r : Core relative density (---)
r : Core relative density including the face sheet (---)
cr : Critical buckling stress for a strut (MPa)
pK : Peak compressive strength of the core (MPa)
flex : Equivalent Flexural shear strength (N/mm2)
y : Yield strength of material (MPa)
xxiii
pK/ y : Normalized compressive strength (---)
pK : Core shear strength (MPa)
pK y : Normalized shear strength (---)
max : Maximum deflection (mm)
: Web angle (deg)
: Inclination angle between core member and face sheet (deg)
Abbreviations
EDM : Electro-discharge machining
PCM : Periodic Cellular Material
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CHAPTER 1INTRODUCTION
1.1 Background
Transport applications frequently need metal structures that are stiff, strong,
tough and light. The choices that best achieve this are often the light alloys [1].
However, the selection of the optimum structure is a difficult task. In general, compact,
lightweight structures that support loads in an efficient and cost-effective way are
valued. Cellular metal structures, that are either stochastic (metal foams) [2, 3] or
periodic (PCMs) [1, 4, 5] are highly attractive types of lightweight structures that are
becoming increasingly utilized in industry [6].
Periodic cellular structures typically display better property profiles than
stochastic cellular materials, at equivalent densities. The last two decades have
witnessed the proposal of PCMs for a wide range of engineering applications such as in
electric sensing and actuation [7, 8], aerospace [9], underwater shock loading [10-12],
heat transfer [13-16], energy absorption [6, 17], and aircraft wings manufacturing [18].
Currently, the most commonly used topology of periodic cellular structures is the
honeycomb. It is found in a variety of applications such as energy absorption, heat
exchange, and as supporting cores in lightweight sandwich panels [15, 19]. The
advantages of honeycomb structures are their high compressive strength and high
bending stiffness [19, 20]. However, other studies have highlighted the potential of
other topologies [4, 12, 21-23]. Micro-truss panel structures are one example. These
structures are created using an interconnected network of solid struts acting as columns
and ties [24-26]. The investigation of the design and performance of these structures has
led to a number of core topologies that can be made according to the application
requirements. Cost-effective methods for micro-truss production have also been
developed, such as casting-based procedures, which permit entire periodic truss
structure components to be produced at scales ranging from millimeters to meters [27].
The overall mechanical properties and performance of assembled micro-truss
sandwich structures are controlled by material properties, structure architecture, e.g.,
core topology [28], and node strength, which is directly correlated to the quality of the
joints [29]. These factors may vary from one application to another depending on the
service conditions. The promising features of micro-truss sandwich structures,
flexibility in manufacturing, relatively inexpensive fabrication processes, and the
increasing demand for these structures in a wide range of applications, have motivated
the current study. The aim of the study is to understand the effects of plastic damage on
the strength and stiffness of micro-truss sandwich structures. With this understanding it
will be possible to design forming techniques and map out tolerable plastic
deformations.
1.2 Thesis Outline
This thesis focuses on the characterization and prediction of deformation
damage response in aluminum micro-truss sandwich structures made of different alloy
grades and subjected to different load modes over wide range of testing conditions.
2
Experimental and theoretical works that have been undertaken in this thesis are
organized in Chapters according to the logical flow of the work, as follows:
Chapter 2 provides an extensive literature review on the principals, theory, and
failure mechanisms of micro-truss sandwiches, along with related subjects such as
concept of plastic damage and forming process. Gaps in the literature and the
corresponding objectives and scope of this work are included in this Chapter.
Chapter 3 provides a detailed description of the research methodology applied in
the current experimental and simulation work. In addition to physical and chemical
characterization of materials, this Chapter describes micro-truss fabrication procedures
by electro-discharging machining (EDM) and brazing along with brazing optimization
using lap-joint joins, simulating brazed nodes in micro-trusses, and mechanical testing
methods (tensile, hardness, compression, and shear). The simulation procedure, which
includes input files, boundary conditions and simulation testing methodologies (for
compression, shear, and bending) are also described in this Chapter.
Chapter 4 presents tensile characterization results of alloys used in this work (as
received AA5083 and AA3003, and annealed AA3003). Tensile results are presented,
compared, and discussed for both grades of alloys over wide ranges of temperature and
strain rate. Tensile results at elevated temperatures were obtained from jump tests.
Calculated mechanical properties, including tensile yield and peak strengths, uniform
and total elongations, in addition to plastic deformation parameters, such as strain
hardening exponent and strain rate sensitivity are provided for both alloys. Accordingly,
Chapters 3 and 4 provide the reader with sufficient information on alloy properties over
a wide range of test conditions, brazing specification and node brazing quality prior
proceeding with testing results of micro-truss sandwiches.
3
Chapter 5 presents compression experimental and simulation results of micro-
truss sandwiches. Emphasis is given to understand the deformation behavior and failure
mode of brazed and machined micro-trusses under different compression conditions.
Experimental results include compressive load tests at room and load/reload tests at
elevated temperatures. Simulation results include load/reload tests at different strain
hardening exponent and strain rate sensitivity at room temperature.
Chapter 6 presents experimental and simulation results for micro-truss
sandwiches in shear and bending. The deformation behavior and mechanical strength
degradation of structures made of different grades of alloys and different node type are
analyzed in this Chapter. Emphasis is given to understand the degradation levels of the
core to provide understanding of the deformation in bending due to the occurrence of
shear.
Chapter 7 presents the conclusions from the current work. Finally, contributions
of the current work to knowledge are presented in Chapter 8.
4
CHAPTER 2LITERATURE REVIEW
Metallic micro-truss sandwich structures have the potential to be used for
supporting structural loads in applications where minimum mass solutions are required.
They offer significant advantages over equivalent mass per unit area monolithic plates
when exposed to certain operational challenges. This chapter presents the fundamentals
of micro-truss structures along with their manufacturing methods, deformation
behavior, failure modes, and applications. Previous experimental, theoretical, and
simulation studies on these structures are discussed and the key research questions
addressed by this thesis highlighted.
2.1 Cellular Materials
2.1.1 General Principles
Periodic cellular materials (PCMs) are characterized by lightness, stiffness,
strength and their multifunctional capabilities [27, 30, 31]. They compare favorably in
their mechanical properties over monolithic plates of equal mass per unit area [32, 33].
PCMs, such as trusses, can be considered as hybrids of space and metal in which beams,
wires or hollow tubes are arranged in a periodic, 2D or 3D architecture [34, 35]. Recent
studies have shown that the resulting architecture can significantly improve the
strength-to-weight and stiffness-to-weight ratios of PCMs over conventional metallic
stochastic structure (foams). For instance, at the same relative density, e.g. 0.1, a foam
structure is less stiff by a factor of 10 than a triangulated lattice [36]. In the case of
foam, a relative density (the relative density is defined as the ratio of the truss volume to
that of the unit cell) of 0.1 means that the solid cell walls occupy 10% of the volume,
whereas for the lattice it means the solid struts in one unit cell occupy 10% of the
volume of that unit cell. The improvement in mechanical properties is due to the fact
that the total material mass is reduced by retaining only that which has a high load-
bearing efficiency [4, 5, 28]. Figure 2.1 illustrates the difference in structures and shear
modulus between stochastic and periodic (lattice) cellular materials (shear modulus
describes material's response to shearing strain represented by the ratio of shear stress to
shear strain).
Figure 2.1: (a) a schematic illustrating the two predominant topologies exhibited by cellular metals, and (b) a comparison of the shear modulus
measured on stochastic closed cell aluminum alloys [4].
6
Figure 2.2 shows the compressive strength of cellular materials and structures
compared with foam. It can be seen that at low relative density, i.e. 4% to 10%, the
pyramidal truss structure strength is comparable to honeycomb and superior to foam.
For these reasons, PCMs "or lattice materials" have become an attractive option for
weight-limited engineering applications, such as panel stiffening in panel constructions
[20].
Figure 2.2: Compressive strength of cellular materials [5].
2.1.2 Description and Comparison of Different Cell Types
PCMs are often incorporated into sandwich structures comprising two face
sheets and a cellular core. With the progress in design and manufacturing techniques,
PCMs are fabricated in different core types depending on the application. The main
types of PCMs are:
7
1. Honeycombs: This topology is typically used in three forms: hexagonal, square
configurations are always superior to textiles over the range of densities examined.
Figure 2.6: Comparison of the compressive stresses at yield as a function of relative density [5]. ( cy is initial yield, y is yield strength, and c is
relative density).
In general, pyramidal truss plate core sandwiches made of aluminum or stainless
steel were found to have a wide design property range as they have an excellent
combination of compression strength and bending rigidity [44]. These properties
provide engineers with more space to maneuver with the design to fit a certain
application without compromising the mechanical strength of the structure. The present
study will focus on micro-truss type structures and explore their mechanical properties
and deformation behavior in different loading modes with a view to understanding the
role of plastic damage.
13
2.2 Micro-Truss Sandwich Structures
2.2.1 Applications of Micro-Truss Sandwich Structures
In general, micro-truss sandwich structure design is application dependent. The
structure design is chosen carefully for each application since the structure properties
are influenced by many factors. Among these factors are: the geometry of the panel,
namely, the apportionment of mass between each face and the core, the core thickness
(or relative density for a fixed face separation), and the core topology. These factors are
also sensitive to the material used to fabricate the structure. The importance of the
material is that its mechanical properties (modulus, yield strength, strain hardening), in
conjunction with the geometry, determine the critical loads, and the failure mechanisms
of core collapse and the face deformation [5, 20, 45-47].
Load supporting micro-truss structures can simultaneously provide excellent
mechanical impact and blast absorption performance [48, 49]. Applications for
underwater blast shock (high-velocity impacts) were tested using cylindrical supports
made of micro-truss sandwich structures to simulate conventional solid structures [6,
45, 50]. The use of these structures is characterized by the reduced momentum transfer
from the water to the panel. This is due to the low inertia of a thin (light) face sheet
supported by a crushable core under an impulsive load [12, 51]. The mechanical
performance of these truss structures partially damaged by underwater blast shock will
be better understood with the aid of results from this thesis. In this work, the
degradation in the mechanical properties of truss structures partially damaged at
different levels of strain will be examined.
There is a growing interest in utilizing micro-truss sandwiches made from PCMs
in shock protection in low-velocity impact, such as component packaging, head impact
protection and vehicle occupant injury prevention during automobile accidents [50].
14
Essential parts of aircrafts [9] and main frames of vehicles [50-54] can be made from
cellular material truss panel structures to improve ride characteristics.
PCM sandwich structures are also becoming popular in the space industry [55].
However, these structures are usually larger than the capacity of carriers transporting
them from earth to orbit. It is then necessary to build the structures in orbit by
combining the components of the structures. For cost reasons, it is important to develop
techniques for constructing large structures in space that minimize in-orbit activities.
Also, it is expected that in many cases the volume of the structural components will
affect the transportation cost. Therefore, it is very important to develop techniques of
packaging the structural components very compactly. To accommodate these two
requirements, 1D and 2D truss structures made from cellular materials and packaged in
a very compact volume are promising option. These materials can be transported easily
and capable of deployment to final configurations in orbit by simple assembling.
Because of their relatively high rigidity, many one- and two dimensional truss structures
have been considered good choices [56].
Micro-truss structures can also incorporate actuators and sensors that are
integrated into the structure. Actuating, sensing, and signal processing elements may be
incorporated into a structure for the purpose of influencing its state or characteristics.
Classical examples of such structures are conventional aircraft wings with articulated
leading- and trailing-edge control surfaces and robotic systems with articulated
manipulators and end effectors. These structures have sensors which might detect
displacements, strains or other mechanical states or properties, electromagnetic states or
properties, temperature or heat flow, or the presence or accumulation of damage [9].
In general, the agreement between UTS of AA5083 obtained in this work and
those reported in the literature is good, as shown in Figure 4.3.
88
Figure 4.3: A comparison between the UTS of AA5083 from the current and previous work [120].
Figures 4.4 and 4.5 illustrate the ambient engineering stress-strain curves of the
two AA3003-1 and 2 sides clad, respectively. The UTS of AA3003-1 side clad varied
from 121 to 113 MPa, and from 117 to 119 MPa for AA3003-2 side clad, when the
strain rate was changed from 2x10-4 to 10-1 s-1. This variation is within errors and not
considered to be significant. The dependency of stress on strain shown in Figures 4.4
and 4.5 agree well with previous observations. A similar insensitivity of stress to strain
rate in the range 10-3 to 0.08 s-1 for fully hardened AA3003 (temper H111) at room
temperature has been reported [109]. Further results that confirm this trend can be also
found in the work of Guo et al. [121] and the work of Tan [122].
[120]
89
0
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5Strain (mm/mm)
Stre
ss (M
Pa)
Specimen 1
Specimen 2
Specimen 30
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5Strain (mm/mm)
Stre
ss (M
Pa)
Specimen 1
Specimen 2
Specimen 3
0
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5Strain (mm/mm)
Stre
ss (M
Pa)
Specimen 1
Specimen 2
Specimen 30
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5Strain (mm/mm)
Stre
ss (M
Pa)
Specimen 1
Specimen 2
Specimen 3
(a) (b)
(c) (d)
Figure 4.4: Ambient engineering stress-strain profiles of AA3003-1 side clad at different strain rates; (a) 2x10-4 s-1, (b) 10-3 s-1, (c) 10-2 s-1, and (d) 10-1 s-1.
90
0
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5
Strain (mm/mm)
Stre
ss (M
Pa)
Specimen 1
Specimen 2
Specimen 3
0
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5
Strain (mm/mm)
Stre
ss (M
Pa)
Specimen 1
Specimen 2
Specimen 3
0
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5Strain (mm/mm)
Stre
ss (M
Pa)
Specimen 1
Specimen 2
Specimen 3
0
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5Strain (mm/mm)
Stre
ss (M
Pa)
Specimen 1
Specimen 2
Specimen 3
(a) (b)
(c) (d)
Figure 4.5: Engineering stress-strain profiles of AA3003-2 side clad at 25 °C and different strain rates; (a) 2x10-4 s-1, (b) 10-3 s-1, (c) 10-2 s-1, and
(d) 10-1 s-1.
The yield strength and UTS of the AA5083 and AA3003 samples are compared
in Figures 4.6 and 4.7. It can be seen that the yield strength of AA5083 is 3-4 times
higher than that of AA3003 (Figure 4.6), whereas tensile strength of the former alloy is
2-3 times higher than that of the latter (Figure 4.7). Also, it is apparent that these
properties of AA5083 appear to vary more than those of AA3003 with increasing strain
rate.
91
0
50
100
150
200
250
0.0001 0.001 0.01 0.1Strain rate (s-1)
Yiel
d st
reng
th (M
Pa)
AA5083AA3003-1 side cladAA3003-2 side clad
0
50
100
150
200
250
300
350
0.0001 0.001 0.01 0.1Strain rate (s-1)
Tens
ile st
reng
th (M
Pa)
AA5083
AA3003-1 side cladAA3003-2 side clad
Figure 4.6: Comparison of ambient 0.02% offset yield strength.
Figure 4.7: Comparison of ambient UTS.
4.3.2 Effect of Temperature
The effect of temperature on the engineering - curves of AA5083 and
AA3003-2 side clad is shown in Figures 4.8 and 4.9, respectively. Although these
curves were performed according to the jump-test method, which creates steps in the
curves, the overall - behavior of the samples is still clear.
92
Figure 4.8: Engineering - profiles of AA5083 at elevated temperatures.Jump-test strain rate range is 2x10-4 to 10-2 s-1.
Figure 4.9: Engineering - profiles of AA3003-2 side clad at elevated temperatures. Jump-test strain rate range is 2x10-4 to 10-2 s-1.
93
The variation of yield strength and UTS with temperature extracted from
Figures 4.8 and 4.9 are plotted in Figures 4.10 and 4.11 for AA5083 and AA3003-1&2
sides clad, respectively, and compared to results from previous studies. The yield
strength and UTS of AA5083 converge at 300 °C. The effect of temperature on yield
strength and UTS of specimens is fairly consistent with that reported in earlier studies
[120, 123]. Similar comments can be made with respect to AA3003 samples (Figure
4.11).
Figure 4.10: Yield strength and UTS of AA5083 at elevated temperaturesand a strain rate of 2x10-4 s-1.
In Figure 4.11, the agreement with values reported by Abedrabbo et al. [109] is
satisfactory taking into account that the sample used in the latter study was a fully
hardened AA3003 (temper H111) tested at strain rate of 8x10-3 s-1. These results show
that the yield strength and UTS of both alloys converge at 400-500 °C.
0
50
100
150
200
250
300
350
0 100 200 300 400 500Temperature (oC)
Tens
ile a
nd y
iled
stre
ngth
(MPa
) Yield-current
Yield-Lloyd
UTS-current
UTS-Clausen et al.
[120]
[123]
94
Figure 4.11: Yield strength and UTS of AA3003-2 side clad at elevated temperatures and strain rate of 2x10-4 s-1.
4.3.3 Strain Hardening Exponent (n) and Strength Hardening
Coefficient (K)
The values of n determined for all alloys by n) [100]
to the uniform strain region of the true stress-strain curves are shown in Figure 4.12.
Figure 4.12a shows the dependency of n as a function of strain rate at room temperature
for both alloys, whereas Figure 4.12b shows the dependency of n on temperature at
strain rate of 2x10-4 s-1. True stress-strain curves used for the calculation of n and K at
different strain rates and temperature are given in Figures 1 through 4 in Appendix IV.
Also, an example of fitting curves is shown in Figure 5 (Appendix IV). Figure 4.12a
shows that n values of AA3003 vary in a linear fashion from 0.26 to 0.28 when the
strain rate is increased from 2x10-4 s-1 to 10-2 s-1. In the case of AA5083, the trend is
different, n changed nonlinearly from 0.32 to 0.19 when the strain rate increased from
2x10-4 to 10-1 s-1.
[109]
95
(a)
(b)
Figure 4.12: The dependency of strain hardening exponent on; (a) strain rate at 25 °C, and (b) temperature at strain rate 2x10-4 s-1.
It is apparent that AA5083 shows higher n values than AA3003 at ambient
temperatures. For temperatures greater than 200 °C the two alloys display similar values
for n, as shown in Figure 4.12b. The trend of n values of AA3003 is found to be
consistent with that obtained by Abedrabbo et al. [109] who reported a value of 0.22 at
25 °C and strain rate of 0.008 s-1, while a later study by Ahmadi et al. [135] reported a
[109]
96
0
100
200
300
400
500
600
700
0 100 200 300 400 500Temperature (oC)
K (M
Pa)
AA5083
AA3003-2 side clad
value of 0.17 for a hardened AA3003 at cross-head speed of 0.017 mm/s (the current
strain rates of 2x10-4 and 10-1 s-1 correspond to cross-head speed of 0.005 and 2.5 mm/s,
respectively).
The value of K in n is plotted in Figure 4.13. A steep decline
in K with increasing temperature is seen with convergence between the two alloys
evident at 400 °C.
Figure 4.13: The dependency of strength hardening coefficient on temperature at a strain rate of 2x10-4 s-1.
4.3.4 Strain Rate Sensitivity Index (m)
The strain rate sensitivity index (m) is plotted in Figure 4.14. The values of m
were obtained using the jump-test method and comparison of curves obtained at
different strain rates. Assuming a power law stress-strain-strain rate equation
( mC. ) where C is a constant. m was determined from the slope of the log-log plots
at room temperature and at various strain rates between 2x10-4 and 10-1 s-1 for a fixed
strain level of 5% (after Benallal et al. [136] and Grytten et al. [137]). The logarithmic
plots are given in Figures 6 through 8 in Appendix IV. The results at elevated
97
temperatures were obtained through the application of equation (3.4). The values shown
in Figure 4.14 display a reasonable agreement between the two techniques [129] and
also compare reasonably well with the literature, further comparison with literature is
given in Tables 1 and 2 in Appendix IV. However, it can be seen in Figure 4.15 that m
values of AA5083 are higher than those of AA3003 at temperatures higher than 300 °C.
(a)
(b)
Figure 4.14: Strain rate sensitivity calculated according to rate jump (2x10-
4 to 10-2 s-) – method I; and from - for different tests (2x10-4 to 10-
2 s-1) – method II; (a) AA5083, and (b) AA3003.
0
0.03
0.06
0.09
0.12
0.15
100 200 300 400 500Temperature (oC)
m
Method I
Method II
Abedrabbo et al.[109]
0
0.1
0.2
0.3
0.4
0.5
100 200 300 400 500
Temperature (oC)
m
Method IMethod IILloyd[120]
98
0
0.1
0.2
0.3
0.4
0 100 200 300 400 500Temperature (oC)
m
AA5083 AA3003-2 side clad
Figure 4.15: A comparison of m for AA5083 and AA3003-2 side clad.
4.3.5 Elongation
The response of elongation to strain rate for AA5083 and AA3003 at 25 °C is
shown in Figures 4.16 and 4.17, respectively. Total elongation to failure (TE) and
uniform elongation (UE) of AA5083 decreased from 18.5% and 16.2% at 2x10-4 s-1 to
7.4% and 6.7% at 10-1 s-1, respectively. By contrast the TE and UE of AA3003 are
relatively insensitive to strain rate at room temperature. The AA3003 values are also
higher than those for AA5083.
99
0
5
10
15
20
0.0001 0.001 0.01 0.1Strain rate (s-1)
Elon
gatio
n (%
)
UE
TE
0
5
10
15
20
25
30
35
40
0.0001 0.001 0.01 0.1Strain rate (S-1)
Elon
gatio
n (%
)
UE - 1 side clad UE - 2 side clad
TE - 1 side clad TE- 2 side clad
Figure 4.16: Effect of strain rate on the elongation of AA5083 at 25 °C.
Figure 4.17: Effect of strain rate on the elongation of AA3003 at 25 °C.
The elongation of these alloys is highly dependent on temperature. Figures 4.18
and 4.19 illustrate the elongation response of AA5083 and AA3003, respectively, at a
strain rate 2x10-4 s-1 and temperature in the range of 25-500 °C. It is noticeable that UE
100
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500Temperature (oC)
Elon
gatio
n (%
)UE
TE
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500
Temperature (oC)
Elon
gatio
n (%
)
UE
TE
decreases at temperatures higher than 100 °C, and diminishes at temperatures higher
than 300 °C.
Figure 4.18: Ductility of AA5083 as a function of temperature at strain rateof 2x10-4 s-1.
Figure 4.19: Ductility of AA3003-2 side clad as a function of temperature at strain rate of 2x10-4 s-1.
101
0
20
40
60
80
100
120
0 0.1 0.2 0.3 0.4Engineering strain (mm/mm)
Engi
neer
ing
stre
ss (M
Pa)
ParentAnealed
0
20
40
60
80
100
120
0 0.1 0.2 0.3 0.4Engineering strain (mm/mm)
Engi
neer
ing
stre
ss (M
Pa)
ParentAnnealed
4.3.6 Annealing
Micro-truss structures studied in this work include machined and brazed trusses.
For the machined structures the properties of the base alloy are still valid for estimating
micro-truss structure properties. Whereas structures assembled by brazing will
experience loss of some of the work hardening inherent in the base alloy due to the
annealing effect. Thus, using the original properties of the base alloy could result in
errors when predicting the brazed micro-truss properties. Accordingly, it is important to
quantify the reduction in mechanical properties of the base alloy after being subjected to
brazing at 605 °C.
Ambient tensile results of AA3003 annealed at 605 °C for ~9 min are shown in
Figures 4.20 and 4.21 for strain rates of 10-3 s-1 and 10-2 s-1, respectively, in addition to
the parent (as-received) AA3003 curves. It can be seen that the yield strengths are
similar, the UTS is lower and that both TE and UE drop considerably for the annealed
material.
(a) (b)
Figure 4.20: Comparison of ambient engineering tensile stress-strain for parent and annealed AA3003 at strain rate of 10-3 s-1; (a) AA3003-1 side
clad, and (b) AA3003-2 side clad.
102
0
20
40
60
80
100
120
Yield-1 sideclad
UTS-1 sideclad
Yield-2 sideclad
UTS-2 sideclad
Stre
ngth
(MPa
)
ParentAnnealed
Figure 4.21: Comparison of ambient engineering tensile stress-strain for parent and annealed AA3003-2 side clad at strain rate of 10-2 s-1.
To quantify the reduction in the properties of annealed specimens, important
parameters were extracted from Figure 4.20 and plotted in Figure 4.22. Annealed
AA3003-1 and 2 sides clad showed reductions of 4% and 9% in their UTS, while their
yield strength was compromised by 3.4% and 0.4%, respectively, compared to their
parent AA3003 specimens.
Figure 4.22: Effect of annealing on the mechanical properties of AA3003at 25 °C and a strain rate of 10-3 s-1.
103
4.4 Discussion
The effect of strain rate on the stress carrying capacity of AA5083 shown in
Figure 4.2 indicates the occurrence of serration, which is consistent with previous
observations on AA5083 [120, 123, 138]. This is due to dynamic strain aging. The
critical strain at which serration occurs in AA5083 depends on strain rate, temperature,
and grain size [123]. In the current test, the combination of low strain rate at room
temperature appears to promote serration. For applications where serration is
intolerable, low strain rates must be avoided when AA5083 specimens are to be
subjected to tensile load at room temperatures. On the other hand, serration in AA3003
specimens was not observed as this alloy is insensitive to strain rate, as shown in
Figures 4.4 and 4.5.
The decrease in the work-hardening capacity with increasing strain rate for
AA5083, observed on UTS variation with strain rate in Figure 4.3, is consistent with the
suggestion of Lloyd and Tangri [139] that dynamic strain aging enhances work
hardening due to the restriction of dynamic recovery mechanisms. The severity of
dynamic strain aging decreases with increasing strain rate, and as dynamic recovery
mechanisms become more efficient, resulting in decreased work hardening. However,
the current results examined the behavior of UTS over a wider quasi-static range of
strain rates than those studied by Lloyd [120] (the strain rate range in the current work
starts at 2x10-4 s-1 compared to 2x10-3 s-1 in Lloyd’s work), thus enabling examining the
behavior of UTS of AA5083 at strain rates smaller than 0.002 s-1. The significance of
the current results is that UTS exhibited a non-linear profile with increasing strain rate,
which in contrast to that observed by Lloyd [120]. This is important as it highlights the
nature of change in UTS with strain rate, and the strain rate at which the UTS is
maximum. On the other hand, Figures 4.4 and 4.5 show that both AA3003-1 and 2 sides
104
clad specimens displayed no evidence of serrations over the entire range of strain rate at
25 °C. Both AA3003-1 and 2 sides clad alloys behaved similarly, as shown in Figures
4.6 and 4.7. It is also clear that the flow stresses are considerably lower in the AA3003
grade compared to AA5083 at all strain rates. The current findings show that the clad
layer has an insignificant effect on the UTS of AA3003. This implies that the current
alloys (AA3003 -1 and 2 sides clad) can be used without compromise in mechanical
strength due to the clad layer.
The UTS and yield strength of AA3003 were examined over an extended range
of temperature, i.e., 25 to 500 °C, compared to the limited temperature ranges (<300 oC)
studied in previous works [109]. On the other hand, Figure 4.12a shows that strain
hardening exponent (n) of AA5083 is found to be sensitive to strain rate change,
whereas both AA3003 specimens exhibited similar dependency of n on strain rate,
where n varied slightly with increasing the strain rate demonstrating a weak dependency
on strain rate. These results suggest that the capacity for strain hardening of AA5083 is
more sensitive to strain rate change than that of AA3003. However, despite the fact the
n for AA5083 is higher than that of AA3003 at low temperatures, n values for both
alloys become nearly equal at temperatures higher than 300 °C This indicates that at
such elevated temperatures, the elasticity of both AA3003 and AA5083 tends to be
similar. It also suggests that such condition compromises the hardness of AA5083
significantly, which is related directly to the degradation in stiffness and strength of the
micro-truss structure at elevated temperatures.
Strain rate sensitivity (m) for all specimens in this work was also obtained over
extended ranges of temperature and strain rate, compared to previous works [109, 120].
For instance, at 25 °C the tests were performed at a strain rate range of 10-4 to 10-1 s-1,
whereas at temperatures >100 °C the tests were performed at a strain rate range of 10-4
105
to 10-2 s-1. These conditions are wider than those applied in Lloyd’s work [120], who
performed tests at a temperature range of 20-400 °C and strain rate of 1.6x10-4 s-1.
Furthermore, in the work of Abedrabbo et al. [109] tensile tests were performed at a
temperature range of 25-260 °C and strain rate of 8x10-3 s-1. Further details on other
works are provided in Tables 1 and 2 in Appendix IV. Clearly, the current conditions
are more comprehensive than those applied in previous work. The difference in m
values of AA5083 and AA3003 at temperatures higher than 300 °C shown in Figure
4.15, is attributed to the difference in alloying elements, e.g. Mg and Fe [120, 140], in
the two alloys as shown in Table 3.1. In general, higher m promotes more uniform
buckling of struts, which decrease stress flow localization, which is a desirable feature
in the forming processes of trusses.
The elongation of AA5083 (Figure 4.16) was limited compared to that of
AA3003 (Figure 4.17). The limitation can be rationalized by the effect of alloy’s
chemical composition. Luo et al. [140] showed that increasing Fe from 0.03% to 0.23%
in AA5083 reduced the ductility significantly. Accordingly, it appears reasonable that
the current alloy with a Fe content of 0.3% exhibits a reduced ductility. Other studies
[124, 141, 142] showed that high elongations can be obtained with AA5083 by reducing
Fe and Si content in the alloy. It is also apparent in Figure 4.17 that both AA3003-1 and
2 sides clad did not differ significantly in their elongation response to strain rate change,
which is in favor of the previous conclusion on the insignificant effect of clad layer on
alloy’s properties. The UE of AA5083 and AA3003, shown in Figures 4.18 and 4.19,
drops with temperature reflecting a drop in work hardening due to increased recovery.
By 300 °C the UE is negligible. The absence of work hardening at higher temperatures
is likely to be significant for pyramidal truss structures, in which plastic buckling is
important.
106
The effect of annealing on stress carrying capacity of AA3003 specimens
(Figures 4.20 and 4.21), resulted in a drop in work hardening capacity. Both UTS and
yield strength are affected negatively by annealing. Perhaps this is due to dissolution of
dispersion strengthening particles [143]. It has been noted in previous studies that
annealing also causes grain growth which can lead to softening. Xing et al. [143] noted
that annealing of AA3003 at 300 °C promoted the appearance of coarse grains raising a
bimodal distribution of the grain size due to the coexistence of fine grains and coarse
grains. The number of coarse grains was found to increase homogenously and gradually
with annealing time. However, increasing the temperature to 400 °C caused a massive
transformation of fine grains to coarse grains. It is worth noting that increasing the grain
size may occur during forming the trusses at elevated temperature. However, the
formation of coarse grains is undesirable in the constructuction materials of trusses, as it
weakens the strength of the material by enhancing localized stress flow.
4.5 Conclusions
Tensile properties for AA5083 and AA3003-1 and 2 sides clad were determined
as a function of temperature (in the range of 25-500 °C and strain rate (in the range of
2x10-4 to 10-2 s-1). The data obtained will be used in the modeling and calculation
sections in subsequent chapters. The main conclusions are:
1. Both AA3003-1 and 2 sides clad alloys were found to be identical in their properties
(yield strength, UTS, uniform elongation, and total elongation), which indicated that
the clad layer does not have a significant effect on AA3003 mechanical properties.
2. While AA5083 showed higher strengths than AA3003, both alloys showed a
convergence of yield strength and UTS at 300 °C
107
3. Uniform elongation also dropped away at 300 °C reflecting a loss of work
hardening capacity.
4. Strain hardening exponent (n) of both alloys was found to decrease linearly with
increasing temperature higher than 100 °C However, high m values provide some
post uniform elongation.
108
CHAPTER 5COMPRESSIVE DEFORMATION OF MICRO-TRUSS SANDWICHES
5.1 Introduction
During manufacturing or application, the micro-truss structure may experience
deformation due to a variety of loading conditions. This chapter deals with the situation
of a compressive load applied normal to the micro-truss sandwich plate. Under an
aggressive compressive load the deformation can take a serious toll on the strength and
stiffness of the structure. Thus, this will affect the mechanical degradation behavior of
the structure as it is subjected to further loading.
The unit cell in a micro-truss sandwich under compression undergoes a plastic
deformation due to yielding or buckling of cell members (struts), as shown in Figure
2.18. The deformation pattern depends largely on the design parameters of the structure
and the material properties. Therefore, the compressive behavior of a micro-truss
sandwich structure represents a research topic of interest, particularly due to the fact
that compressive loading provides insights to understand the compressive behavior of
the core, which is of interest for understanding forming process of micro-truss
(bending).
It is of interest to investigate the compressive deformation behavior and
estimate the degradation levels of mechanical properties of these structures after plastic
loading. This will lead to a better understanding of the response of these structures in
applications where reloading a partially damaged structures is possible. This chapter
presents the experimental and simulation results of brazed AA3003-2 side clad EDM
and machined AA5083 EDM pyramidal micro-truss structures subjected to a wide
range of compression loading test conditions, e.g. varied temperature, and strain. Finite
element (FE) simulation was performed to predict the deformation profile and
degradation level of the core under the effect of different mechanical and material
The compressive loading work of this chapter has two major sections:
experimental and simulation. In the experimental work, compressive tests were
conducted at room and elevated temperatures. This enabled examination of the effect of
temperature on the compressive deformation behavior of the structures.
Care was taken to ensure that the strain rates used in these experiments were in
the quasi-static region to avoid flow localisation [144]. Room temperature loading tests
were performed on the EDM AA5083 structure at strain rate of 10-4 s-1. The tests at
elevated temperature were conducted with EDM AA5083 and brazed AA3003
structures at strain rate of 10-2 s-1 in load/reload conditions. This strain rate was
necessary to reduce the total time involved in testing all samples to a manageable level.
The different strain rates mean that the results cannot be compared directly but by
selecting strain rates in the quasi-static region it is expected that a comparison of trends
in the resulting curves would be valid.
110
An electrical heater was used for preheating the samples. Tests were allowed to
continue to plastic strain levels of ~ 6, 11, and 17%, in the cycle of load/reload.
A parametric study by FE simulation was carried out to explore the effect of
material constituents and plasticity parameters, i.e. n and m on the compressive
deformation behavior of the structure. The Hollomon equation was used to obtain
values for n from the experimental data. These values were used for analytic
predictions. The Ludwick equation was used for the FE parametric study, due to the
standard approach used in Abaqus. This does not affect the conclusions of the
parametric study.
The simulation work included running load/reload tests at room temperature and
a maximum plastic strain level of 18% (corresponds to 3 mm displacement), according
to the boundary condition of Figure 3.24. The simulations were conducted at strain
levels of ~6%, 11%, and 17%, using experimental tensile data of AA5083 and AA3003-
2 sides clad measured at strain rate of 10-2 s-1. The degradation of stiffness and strength
obtained from these tests were then compared with experimental results. Further tests
were also conducted at different n, and different m. The simulations at different n and m
were carried out using tensile data of AA5083 at strain rate of 2x10-4s-1 described in
section 3.5.3.2, and strain levels of 5.5%, 10%, and 15.63% at each value of n and m.
Values of n and m were varied from 0.05 to 0.4 and from 0.1 to 0.4, respectively.
Simulations were carried out at a maximum reload strain level of 18%, which was
maintained constant throughout all simulations. Reload strains up to a maximum of
20% were used. This ensured that no contact between face sheets and buckled struts
should be occurred during the tests. At each level of reload strain, e.g. 5.5 %, yield
occurs at a slightly lower load than the strength at the point of unloading. This lower
load bearing capacity is attributed to the fact that loading the structure causes loss of
’
111
some of the work hardening due to straining the structure. When the structure is
unloaded some of the lost work hardening is recovered, while the lost work hardening
causes the load bearing capacity of the structure in the next loading to be lower than that
in the previous cycle. Calculating this loss is useful in estimating the mechanical
degradation performance of the micro-truss structures under different levels of reload
strains. In addition, these reload strains are widely used in previous studies [3, 46, 63].
Further details on the methodology of compressive tests are shown in the block
diagram presented in Figure 5.1.
112
Figure 5.1: Diagram of compressive tests methodology.
Experimental compressive test
Load/reload test Load test
25 oC, 10-4 s-1, EDM AA5083
25, 300, and 500 oC, 10-2 s-1, EDM AA5083 and brazed
AA3003
Compressive experiments
Simulation compressive load/reload test at 25 oC
Comparison with experiment
Using tensile data of AA5083 and AA3003-2
side clad at 10-2 s-1
Different n Different m
Using tensile data of AA5083 at 2x10-4 s-1
113
5.3 Results
5.3.1 Basic Load-Displacement Behavior
The ambient temperature compressive responses of two AA5083 truss structure
compressed at 10-4 s-1 are shown in Figure 5.2. The responses displayed characteristics
typical of truss sandwich structures. After some initial bedding-in, the structure
responded elastically over a compressive displacement of ~0.3 mm. After this stage of
elastic response, a gradual core yield occurred followed by a peak compressive strength
at ~2030 N. This corresponded to the initiation of strut buckling and plastic hinge
formation. Photographs in Figure 5.2 show the state of the truss compressed to different
strains.
Figure 5.2: Compressive load-displacement of 2x2 AA5083 structure compressed at 25 °C and strain rate of 10-4 s-1.
Continuing loading at a constant strain rate resulted in core softening, which was
accompanied by a significant decrease in load carrying capacity of the structure. Core
softening occurred over displacements from 0.3 to 3 mm. Upon further loading the load-
114
displacement curves displayed a sudden sharp increase in load carrying capacity due to
the contact between buckled members and face sheets. This has been termed “core
densification” [55]. No evidence of node failure or member fracture was observed
during the test. In addition, no buckling or yielding was observed in the face sheets. The
difference between the responses of the two specimens is attributed to imperfections in
geometry. A summary of the design parameters used for the AA5083 structure used for
these tests (Figure 5.1) is given in Table 5.1. Also shown are the peak loads and the
corresponding stresses, obtained by dividing the load by the plate area, Ap, which has a
value of 3481 mm2 (values of unit cell dimensions are presented in Table 3.2).
It is worth noting that most of the literature on micro-truss compression
describes the initial elastic response of the structure as linear [32, 51, 62]. A magnified
plot of this section for specimen #1 is given in Figure 5.3. The figure indicates that the
near linear region ends at around 1200 N due to the occurrence of plastic yielding (an
offset was used which is a normal practice).
.
Figure 5.3: A magnified bedding-in section of Figure 5.2.
115
It is well recognized that pyramidal micro-truss structures fail typically by
inelastic buckling when loaded in compression [31, 63]. The peak compressive load
(FpK) for a sandwich of 4 unit cells was predicted. Equation (5.1) shows the analytical
expression used in the theoretical prediction of the peak compressive strength [31, 63].
r2
crpK .sin (5.1)
pK is the p cr is the critical buckling stress for a strut,
is the strut inclination angle (defined in Figure 3.2) and r is the relative density.
In order to predict the failure by inelastic buckling, the critical stress, cr, is
calculated using the following equation [28, 55, 69, 145]:
2cs
t22
cr LAIEk
(5.2)
where Et is the Shanley-Engesser tangent modulus (the slope dd of the stress-strain
curve), As is the cross-sectional area of the strut, and I is the second area moment of
inertia of the strut (which equals to a3 b/48). The calculation of I is given in Appendix
III for a rhombic cross section. The k value was set at 2 [55, 69]. Clearly, this
calculation requires prior determination of Et. As noted in section 4.3.3 in Chapter 4, the
stress-strain curves in the present study follow a power law ( n.K ), the slope
is 1-Kdd nn . This can be written in term of stress as
nn
n1
KK
dd . Thus,
equation (5.2) becomes:
116
nn Ikn 2
cs
22
cr LAK (5.3)
This provides a convenient expression to use for the prediction of cr . The strength
parameters of pK , FpK, and the geometrical values used in the prediction of these
parameters are given in Table 5.1. This prediction agrees quite well with measured
values of the peak load in Table 5.1 (2030 N is the average of the peaks of two
specimens, i.e.1922 and 2139 N).
Table 5.1: Theoretical prediction and experimental averaged values of Strength parameters for AA5083 structure with r = 0.019, As = 2.436, and I = 0.287 compressed at 25 °C and strain rate of 10-4 s-1.
Structure FpK
(N)
pK
(MPa)
K*
(MPa)
n*
Theoretical 2023 0.581 600 0.31
Experimental 2030 0.583
* Values are taken from Figures 4.12a and 4.13
5.3.2 Influence of Temperature on Loads
Theoretical predictions of inelastic buckling were made using Equation (5.3)
with values from Tables 5.2 and 5.3. Values of geometrical dimensions for calculating
As and I in Table 5.2 were taken from Table 3.2, and y were taken from
Figures 4.2c and 4.5c at 25 °C and Figures 4.8 and 4.9 at 300 °C and 500 °C for
AA5083 and AA3003 structures, respectively, whereas values of K and n in Table 5.3
were taken from Figures 9 through 14 in Appendix IV. Further details on the calculation
of geometrical dimensions are given in Appendix III.
117
Table 5.2: Geometrical values used in the theoretical predictions.
Structure r As
(mm2)
I
(mm4)
AA5083 0.019 2.436 0.287
AA3003 0.019 2.567 0.335
Table 5.3: Material properties and values of stress used in the theoretical prediction of load/reload compressive response at elevated temperatures
and strain rate of 10-2 s-1.
Temperature
(°C)
y
(MPa)
K
(MPa)
n pK
(MPa)
FpK
(N)
AA5083
25 103 675 0.3 0.683 2377
300 50 76 0.053 0.33 1138
500 10.4 12.43 0.02 0.067 234.5
AA3003-2side clad
25 45 217 0.25 0.297 1075
300 23 30 0.032 0.153 554
500 11 13.5 0.023 0.073 262
The measured compressive load versus displacement curves of AA5083
structures are shown in Figures 5.4 through 5.6 for test temperatures of 25, 300, and 500
°C respectively. Those for AA3003 structures are plotted in Figures 5.7 through 5.9,
respectively. The compression was conducted at a constant strain rate of 10-2 s-1. The
results were also compared to the theoretical predictions in these Figures.
118
Figure 5.4: Compressive load/reload cycles of AA5083 structure at 25 °Cand 10-2 s-1. Photographs are for specimen #1. Prediction shown for K= 675
and n= 0.3 (from Chapter 4) using equation (5.3).
Figure 5.5: Compressive load/reload cycles of AA5083 structure at 300 °C and 10-2 s-1. Photographs are for specimen #1. Prediction shown for
K=76 and n=0.053 (from Chapter 4) using equation (5.3).
119
Figure 5.6: Compressive load/reload cycles of AA5083 structure at 500 °Cand 10-2 s-1. Photographs are for specimen #2. Prediction shown for
K=12.43 and n=0.02 (from Chapter 4) using equation (5.3).
Figure 5.7: Compressive load/reload cycles of AA3003 micro-truss structure at 25 °C and 10-2 s-1. Photographs are for specimen #1. Prediction
shown for K=217 and n=0.25 (from Chapter 4) using equation (5.3).
120
Figure 5.8: Compressive load/reload cycles of AA3003 micro-truss structure at 300 °C and 10-2 s-1. Photographs are for specimen #1. Prediction shown for
K=30 and n=0.032 (from Chapter 4) using equation (5.3).
Figure 5.9: Compressive load/reload cycles of AA3003 micro-truss structure at 500 °C and 10-2 s-1. Photographs are for specimen #1. Prediction shown for
K=13.5 and n=0.023 (from Chapter 4) using equation (5.3).
121
In these Figures, a plastic compressive strain of 20% was taken as the maximum
allowable level where no contact between buckled struts and face sheets occurs. The
load/reload cycles were performed at plastic strains of 6, 11, and 17%. The peak load
decreased from 2365 N at 25 °C to 231 N at 500 °C for AA5083, which represents a
reduction of 90%.
Interestingly, the photographs of the deformed AA5083 structures suggest that
the higher the temperature is the higher the displacement (or
formed. For instance, at 300 °C and 500 °C (Figures 5.5 and 5.6 respectively), the hinge
°C (Figure 5.4).
This possibly arises due to the effect of temperature on Young’s modulus. However, the
effect was not seen in the AA3003 data. It can be seen that the predictions of loads
required for inelastic buckling agree reasonably well with the measurements.
In the case of AA3003 structure, higher temperatures promoted the formation of
a hinge at lower displacements, for instance, at 25 °C (Figure 5.7) the hinge formation
was initiated at a displacement of 0.49 mm
°C (Figure 5.9). No node failure was observed in
any structures at any of the temperatures tested, which indicates that the brazing
condition applied in this work was sufficient to produce a high quality brazed nodes.
Clearly, the load carrying capacities of both AA5083 and AA3003 structures were
highly compromised at elevated temperatures, as would be expected.
Finally, the results in Figures 5.5 through 5.9 show that the severity of the
hysteresis between the unloading and loading curves appears to increase with increasing
strain and temperature. The hysteresis is likely to be due to the role of the elastic
recovery and residual strain behavior during load/reload, and the behavior of the tangent
122
modulus during unloading which corresponds to the instantaneous stiffness of the
structure. Thus the hysteresis is a function of material properties, design specifications
and operating conditions (temperature and strain). The sudden change in slopes at low
loading forces is attributed to the drop in stiffness of the structure. Accordingly, at
higher temperature and strain, the drop in stiffness is more significant leading to a more
pronounced hysteresis due to higher work softening. Therefore, the evolution of
hysteresis is not due to the experimental setups.
The dependency of peak load carrying capacity on temperature for both
structures is compared in Figure 5.10. It can be seen that despite the mechanical
superiority of the AA5083 structure compared to the AA3003 structure at room
temperature, both structures suffered a significant drop in their peak load, reaching a
carrying capacity of only ~232 N at 500 °C. This is consistent with the tensile results
presented in Chapter 4.
Figure 5.10: The effect of temperature on peak load carrying capacity of AA5083 and AA3003 micro-trusses.
123
It can be noted on the photographs of Figures 5.4 through 5.6 for AA5083, and
that the shape of
buckled struts in both structures shifted from sharper localized bending to a slightly
more uniform hinge with increasing temperatures. This may be due to the effect of m,
which is higher at 300 °C and 500 °C than at room temperature (Figure 4.15). Higher
values of m tend to suppress flow localization but the effect here is a subtle one.
5.3.3 Effect of Plastic Damage on Stiffness
The elastic stiffness of the micro-truss structures undergoing compression was
analysed in terms of normalized compressive stiffness (the ratio of core stiffness
modulus, Ec /Young’s modulus of the material, Es), at the same temperature. The core
stiffness was determined from the ratio of yield strength to strain at yield (obtained from
equation 5.4), where the yield strength is defined as the compressive load at yield
divided by the plate area. Fy was calculated from the load-displacement data of the
trusses by selecting the force corresponding to a plastic strain of ~0.002. Young’s
modulus of the material was obtained from [146] and the values are given in Table 5.4.
Table 5.4: Normalized compressive stiffness for load stage.
In general, the current structures showed a good strength rather than stiffness
(better resistance to degradation in strength than to stiffness), suggesting that the current
design appears suitable for high-strength applications. These findings improved the
understanding of what parameters and properties are essential in the design of a
pyramidal micro-truss structure from aluminum alloys.
5.5.3 Implications
The main objective of this chapter is to understand the effect of plastic damage
on the mechanical properties and compressive behavior of micro-truss structures made
of two different grades of alloy, i.e. AA5083 and AA3003, with two types of node, i.e.
identical (non-brazed) and brazed. The significance of this work is to determine the
maximum plastic compressive strain that can be applied on a micro-truss structure and
the corresponding degradation in mechanical properties (stiffness and strength) of the
structure during load and reload cycles. This work predicted the maximum strain level
that should not be exceeded in order to maintain superior performance of the micro-
truss structure over that of a foam structure of an equivalent density. This also enabled a
better understanding of how the features of the micro-truss structure, e.g. identical or
brazed nodes, material properties, design, and operating conditions, e.g. temperature and
strain, influenced the degradation behavior of stiffness and strength.
It was found that the experimentally-measured normalized stiffness (Es/Ec) of
AA5083 structure is less sensitive to temperature (Figure 5.11a) than AA3003 structure
(Figure 5.12a). Both structures showed a relatively insensitive Es/Ec to reload strain
(Figure 5.20). In general, experimental results showed that straining AA5083 and
AA3003 structures beyond ~6% would result in a performance comparable or inferior to
that of a foam. Thus, the current structures should not be subjected to strains higher than
145
the specified values at room temperature in order to maintain the gain obtained from
using truss structures over foams.
The behavior pK y) (Figure 5.23) obtained from
simulation differs from that of Es/Ec (Figure 5.22). At room temperature, both AA5083
pK y compared to that of a foam structure
even when the trusses were deformed at a plastic strain of 17%. On the other hand, a
strain of 11% was the maximum at which the structures can be compressed in reload
without loosing their stiffness superiority over the foam. The effect of plastic strain was
found to be more critical than that of both n and m in influencing the degradation of
Es/Ec pK y. With strain level being more influential than alloy grade, these results
imply that the design of the trusses is application-oriented; a micro-truss structure that is
good for applications where strength is of interest is not necessarily going to be suitable
for applications where high stiffness is required. Finally, it is worth concluding that
both structures designed in the current work are more favoured for applications
that require strength rather than stiffness regardless of Al alloy grade, taking into
account that the structures in the application should not be strained to more than
16% to maintain superiority over a foam panel.
5.6 Conclusions
This work investigated the deformation behavior of AA5083 and AA3003
pyramidal micro-truss structures experimentally and theoretically. Load/reload tests
were performed to examine the degradation pattern of strength and stiffness of a pre-
deformed structure. The degradation in the mechanical properties of the structures were
also studied over a wide range of temperature, and plastic strain to understand the
combined effect of these parameters on the degradation level of the properties of the
146
structure. Prior to concluding remarks, it is important to address that brazed nodes
demonstrated high strength, with no node failure was observed in any tests. This
indicated that the brazing condition and method applied in this work are reliable for
assembling micro-truss structures. A number of conclusions can be addressed:
1. Increasing the temperature from 25 °C to 500 °C did not change the failure
mechanism of both AA5083 and AA3003 structures, which were found to fail by
inelastic buckling of the struts.
2. The degradation in the peak strength and stiffness of both structures were highly
sensitive to temperature. The higher the temperature is the lower the strength and
stiffness. At 500 °C, the degradation in strength can reach 10% and 25% of the
corresponding values at 25 °C, and 1% and 3.8% of the stiffness at 25 °C for
AA5083 and AA3003, respectively.
3. The degradation of stiffness with plastic strain was found to be relatively similar for
both AA5083 and AA3003 structures. Both micro-trusses were capable of
outperforming the foam, while their stiffness approached that of the foam when they
were deformed at strains up to 6%. However, the effect of strain on stiffness was
found to be less pronounced than that of temperature. On the other hand, the
strength of these structures was superior to that of a foam even when these micro-
trusses were strained up to 17%. Considering this performance, these micro-trusses
are superior for applications demanding high strength.
4. Finite element modeling revealed that both n and m have a limited effect on the
degradation of strength and stiffness of the pyramidal micro-truss structures made of
aluminum alloy. In addition, the stiffness of the structure was more sensitive to
strain than in the case of strength which was less sensitive to strain over the entire
ranges of n and m.
147
CHAPTER 6SHEAR AND BENDING DEFORMATION OF MICRO-TRUSS SANDWICHES
6.1 Introduction
Some applications, such as automotive and aircraft industries, require curved
micro-truss structures. The forming of micro-truss sandwiches into such shapes will
invariably subject the structure to a bending stress where the most highly stressed
trusses can experience either tension or compression, potentially resulting in strut
buckling. However, the generated bending moment distribution may also promote
transverse shear loading. According to previous studies [2, 31] the shear loading can
result in core collapse, thus this type of failure mechanism can dominate the competing
failure modes of a sandwich truss structure in bending [147]. In addition, the partially
deformed structure will suffer degradation in its stiffness and strength to weight ratios.
Similar effects can also be encountered during unintentional damage.
In shear dominated buckling the cross-sections of the core will not undergo
significant rotations, however in bending-governed buckling the cross-sections of the
core rotate and remain approximately perpendicular to neutral axis of bending of the
column [83]. Accordingly, the core of the sandwich must possess adequate flexural
strength to withstand the conditions of the forming process.
This chapter explores the shear deformation behavior and node strength of
AA5083 and AA3003 micro-truss structures in load and reload tests using both
experiments and simulations. The results from shear tests will be used to develop
understanding of the bending deformation behavior of AA5083 micro-truss structures.
Bending deformation is studied using 4-point bending test simulations to model the
mechanism of collapse during forming in more detail. Emphasis is given to
understanding the degradation levels of the equivalent flexural core shear strength of
these structures with increasing sandwich curvature.
6.2 Methodology
The methodology of the work included in this chapter is shown in Figure 6.1.
The work has two major sections: experimental and simulation. All tests were carried
out at room temperature. In the experimental work, load/reload shear tests were
conducted with 2x2 unit cells of EDM AA5083 and brazed AA3003 structures at a
strain rate of 8x10-3 s-1 (this strain rate was selected to be within the quasi-static range
of strain rate applied in Chapter 5). The tests were conducted separately, i.e. load stage
followed by a separate reload stage. The maximum limits of strain level applied in load
stage (no contact between buckled struts and face sheets was allowed) were set to be up
to 11% and 9% for EDM AA5083 and brazed AA3003 structures respectively, and 20%
in reload stage. A pair of specimens was fixed in the grips as shown in Figure 3.19.
149
Figure 6.1: Diagram of shear and bending tests methodology. All tests were carried out at 25 °C. Test conditions shown for experimental shear, simulation shear and bending are taken from sections 3.4.4.1, 3.5.7, and 3.5.6, respectively.
Simulation tests
Shear and bending tests
Experimental tests
Tensile data of AA5083 at 2x10-4 s-1
were used in the input file.Strains during initial loading = 6% and 11% for EDM AA5083, and 7% and 9% for brazed AA3003.Strain during reloading = 20%.
Shear tests.Load/reload stages carried out separately.Strain rate of 8x10-3 s-1 (both stages).
Shear tests.Load/reload stages.
4-point bending tests.Load stage only.
Using EDM AA5083 and brazed AA3003 structures
Using EDM AA5083 and EDM (non brazed) AA3003
Input files for AA5083 and AA3003 were tensile data in Table 6.1 for
measurement at strain rate of 8x10-3 s-1
Using EDM AA5083 only, with 3x3 unit cells at three ts:tf ratios
of 0.5, 1, and 2.
Maximum strain during loading = 5% (no contact between buckled struts and
face sheets).
Strains during initial loading = 6% and 11% for EDM AA5083, and 7% and 9% for EDM AA3003.Strain during reloading = 18%.
150
Simulation work included two major parts, namely shear and 4-point bending
tests. Shear tests were conducted with a single unit cell of EDM AA5083 and non-
y, K, and n) given in Table 6.1
for measurements at strain rate of 8x10-3 s-1. These tensile data were selected to obtain a
consistent comparison with results from shear experiments. The aim was to predict 1st
order trends not more subtle node effects so the brazed node material was not included
in the simulations. The strain-stress curve generated as an input file for the simulations
was obtained by interpolating data presented in Appendix IV. Strain levels applied in
load/reload shear simulations corresponded to those of experiments at equivalent r for
both EDM structures. These strains were 6 and 11% for EDM AA5083 structure, and 7
and 9% for EDM AA3003 structures in load and 20% in reload.
Table 6.1: Interpolated tensile data used in input files of shear simulationsat strain rate of 8x10-3s-1 and 25 °C.
Simulations of 4-point bending were conducted for a step load only using tensile
data of AA5083 at 25 °C and strain rate of 2x10-4 s-1 (such slow strain rate is
recommended for forming process of aluminum alloys [144]). The micro-truss structure
consisted of 3x3 unit cells. The tests were carried out at three ts:tf ratios of 0.5, 1, and 2
at a maximum strain level of 5% with no contact between buckled struts and face
sheets.
Alloy y
(MPa)
K n
AA5083 118.8 570.3 0.23
AA3003 45.5 220 0.25
151
0
200
400
600
800
1000
1200
1400
1600
0 0.5 1 1.5 2 2.5 3Displacement, mm
Shea
r loa
d, N
20% strain-reload(presrained at 6%)
20% strain-reload(prestrained at 11%)
0
200
400
600
800
1000
1200
1400
1600
0 0.5 1 1.5 2 2.5 3Displacement, mm
Shea
r loa
d, N
6% strain-load
11% strain-load
6.3 Results
6.3.1 Basic Measured Load-Displacement Behavior
The experimentally measured results of shear load versus displacement for
AA5083 and AA3003 structures tested at strain rate of 8x10-3 s-1 and 25 °C are plotted
in Figures 6.2 and 6.3, respectively. The results represent the averaged values obtained
from shearing two specimens simultaneously (according to shear grips setup shown in
Figure 3.19).
(a)
(b)
Figure 6.2: Shear-displacement curve of AA5083 structure with 2x2 unit cells at 25 °C and 8x10-3 s-1; (a) load and (b) reload.
152
In the loading stage for the AA5083 structure (Figure 6.2a), a nearly linear initial
loading is seen, followed by core plastic yielding. The cores continued to support the
load through a gradual strain hardening until an average peak shear force, FpK, of 1552
N was reached. It is noticeable in the images that the load orientation at strain 6%
places two adjacent unit cell struts in compression (recognized by the formation of
hinges) and the other two in tension (recognized by the straight shape). The images
show that struts in compression experienced buckling. Continuing straining to 11%
caused further buckling of those struts in compression.
Figure 6.2b shows the reload stage for pre-strained structures (from Figure 6.2a).
A partially deformed structure at 6% and pre-strained to 20% exhibited FpK of 1219 N,
representing a reduction of ~23% compared to its corresponding initial FpK in load
stage, whereas that pre-strained at 11% and strained to 20% displayed FpK of 909 N,
representing a reduction of 40.2%.
After reaching the peak shear load, continuing loading in the reload stage
produced a load plateau. This reflects the load bearing capacity of the struts deforming
in tension. The increasing load observed on the curve is attributed to the contact of
buckled struts and face sheet, as shown in the corresponding images. However, it can be
seen that increasing the strain of the first load from 6% to 11% has shortened the
plateau region, which has resulted from a faster collapse of the truss. Also, it can be
seen that increasing the strain level of the first loading decreased the slope of the initial
linear region upon reloading. This is an indication of loss of stiffness.
Figure 6.3 illustrates the performance of the AA3003 structure in a shear test. An
average FpK of 583 N structure was reached by AA3003 structures strained to 7% and
9% (Figure 6.3a). Straining to 20% in reload resulted in reductions of 16.6% and 24.8%
in the FpK of these structures with respect to their initial FpK in load stage (Figure 6.3b).
153
0
100
200
300
400
500
600
700
0 0.5 1 1.5 2 2.5 3
Displacement, mm
Shea
r Loa
d, N
20% strain-reload(prestrained at 7%)
20% strain-reload(prestrained at 9%)
0
100
200
300
400
500
600
700
0 0.5 1 1.5 2 2.5 3Displacement, mm
Shea
r Loa
d, N
7% strain-load
9% strain-load
(a)
(b)
Figure 6.3: Shear-displacement curve of AA3003 structure with 2x2 unit cells at 25 °C and 8x10-3 s-1; (a) load and (b) reload.
154
A comparison between the performances of AA5083 and AA3003 structures
reveals that the shear FpK of AA3003 is about one-third of that for AA5083 structure in
the loading stage. It is worth noting in images of these structures that only half of the
struts are buckled (the other half are in tension) suggesting that only half of the struts
carried load. Node rupture was not observed in either of AA5083 and AA3003
structures over the entire range of strain. In general, it can be realized that the shear
force carrying capacity of AA5083 structure appears superior to that of AA3003
structure, with much less degradation rate in its shear strength compared to AA3003
structure.
The performance of the brazed nodes encouraged performing simulations with
non-brazed AA3003 structure which will enable assessing the role of different node
type on the deformation behavior of structures made of similar alloy (AA3003). In
addition, simulations with AA5083 will enable the comparison of performance of
structures made of different alloy grades.
6.4 Simulations
6.4.1 Basic Load-Displacement Behavior in Shear
The general goal of this section is to study the effect of plastic strain on the
performance of the truss core in structures with identical nodes (non brazed) and similar
truss geometry but different alloy grades, i.e. AA5083 and AA3003. Thus, different
mechanical properties were used as input files. The simulated truss core load/reload
shear curves are presented in Figure 6.4. It can be seen that FpK of AA5083 (Figure
6.4a) is three times higher than that of AA3003 (Figure 6.4b), which is fairly consistent
with experimental results (Figures 6.2 and 6.3).
155
(a)
(b)
Figure 6.4: Simulation shear results of load-displacement curves of pyramidal micro-truss structure of 2x2 unit cells with r of 0.019 at 25 °C
and strain rate of 8x10-3 s-1; (a) AA5083 and (b) AA3003.
Strut buckling behavior was similar for cores made of each material, and the
buckling intensity increased with increasing shearing strain. Comparison between
simulation images (Figure 6.4a and 6.4b) and experimental results (Figure 6.2b and
6.3b) for AA5083 and AA3003 respectively indicates that there is reasonable qualitative
156
agreement of deformation behavior between simulation and experiment. Quantitatively
though the struts exhibited a more intense buckling in experiment than in simulation. It
can be also seen that the buckling shape predicted by simulation is more uniform than
that observed in the experiments.
6.4.2 Basic Load-Displacement Behavior in 4-Point Bending
The general goal of this section is to study the effect of plastic strain on the
bending behaviour of the truss core over the three ratios of ts:tf (Table 3.5). The load-
bending behaviour of AA5083 structures designed with ts:tf ratios varied from 0.5 (thick
face sheet) to 2 (thin face sheet) for constant span length (L=72 mm) and core with 3x3
unit cells is presented in Figure 6.5. This design ratio is of interest as it highlights which
geometrical parameter (face sheet thickness or strut thickness) is more important in
determining the failure mechanism of the structure. In the case of ts:tf ratio of 2, the
structure undergoes noticeable plastic yielding prior reaching the maximum load,
followed by softening.
157
0
10
20
30
40
50
60
0 0.5 1 1.5 2 2.5 3 3.5Deflection, mm
F, N
0.00010.0010.0016
0
100
200
300
400
500
0 0.5 1 1.5 2 2.5 3 3.5Deflection, mm
F, N
0.0010.00160.00240.006
0
10
20
30
40
50
60
70
0 0.5 1 1.5 2 2.5 3 3.5Deflection, mm
F, N
0.00010.00040.0010.00160.0024
Figure 6.5: Load-deflection curves of AA5083 structure with 3x3 unit cells in 4-point bending at different r . L=72 mm at 25 °C and strain rate of
2x10-4 s-1.
ts:tf =0.5
ts:tf =1
ts:tf =2
158
For a ts:tf ratio of 0.5 the peak load carrying capacity, FpK, increased slightly
with increasing r , e.g. from ~31 N at r of 0.001 to 55 N at r of 0.0016 (Figure 6.5).
When the ts:tf ratio increased to 1, FpK increased from 16 N to 50 N at the same values
of relative density. This reveals that increasing ts:tf ratio from 0.5 to 1 (thinner face
sheet) resulted in lower FpK, for example lower by ~11% at r of 1.6x10-3. The
consistency of decreasing Fpk with increasing ts:tf ratio is attributed mainly to the role of
face sheet thickness. A thick face sheet provides extra resistance to the initial bending
moment, and thus enhances the stiffness of the structure. Accordingly, the thinner the
face sheet is, the lower the resistance of the structure to the load as the resistance in this
case is contributed solely by the core.
The distribution of force and the resulting bending moment throughout the
structure in 4-point bending were also enhanced by the thick face sheet. Bending
resulted in more uniform deformation when the face sheet was thicker with the
occurrence of identical buckling on both edges of the structure while maintaining a
straight upper face sheet. Further increase in ts:tf ratio to 2 promoted a 41% reduction in
FpK. At this ratio, the face sheet is significantly thin, and hence its role in the
deformation of the structure is negligible.
Structures with low r , e.g. r < ~10-3, exhibited a significantly low FpK at ts:tf
of 0.5 and 2. These results suggest that the higher the r is, the higher the bending
resistance of the structure. Increasing ts:tf ratio also resulted in a larger deflection at
which the peak load was attained.
It can be also seen that changing the ts:tf ratio resulted in different behavior over
the softening region. The higher the ts:tf ratio, the more gradual and uniform the
degradation in the load carrying capacity, i.e. a smaller drop in strain hardening. It must
be noted here that the concave shape of the softening region in the structure with ts:tf of
159
0.5 (and slightly at ts:tf of 1) is an undesirable feature as it is correlated to faster
degradation in stiffness and strength. These results indicate that ts:tf ratio is a critical
design parameter that has a strong influence on the bending carrying capacity of the
structure.
The corresponding images in Figure 6.5 indicate that the structure failed by core
buckling regardless of ts:tf ratio. This is attributed to the fact the trusses that experienced
the largest stresses are in compression, on the edges of the structures. These trusses
buckled plastically, resulting in large, inelastic shear strains, causing plastic hinges at
the centre of the struts. On the other hand, the inner trusses (in tension) of the same unit
cells stretched and sustained the load at essentially fixed strain. Clearly, the 4-point
bending model could describe the deformation progress reasonably, demonstrating that
the thinner the face sheet is the lower the strength of the structure. In addition, it
produced an even buckling phenomenon.
It can be concluded that the design parameters of ts:tf ratio and r have a
pronounced effect on FpK of AA5083 pyramidal micro-truss structures. Furthermore, to
understand the core deformation independently of the face sheet, a ts:tf ratio of 2 can be
used as the face sheet at this ratio has a negligible effect on deformation. The higher the
ts:tf ratio is the lower the FpK of the structure at similar r , while the higher the r is the
higher the FpK at similar ts:tf ratio.
6.5 Discussion
6.5.1 Analytical Prediction of Peak Loads
Analytical prediction of mechanical strength was calculated in this section to
verify the experimental and simulation results for further assessment of the current
micro-truss structures. FpK values were predicted analytically for comparison with
160
experimental and simulation shear results, whereas cr was predicted for comparison
with simulation bending results. The calculation methods and results are presented next.
For shear analytical predictions for comparison with experimental and
simulation results, the core shear strength, pK , of AA5083 and AA3003 micro-trusses
subjected to failure by inelastic buckling was calculated using equation (6.1) [63]. The
cr, in this equation was calculated using equation (6.2). The
calculation results were then compared with those from experiment and simulation. In
the calculation of FpK (FpK= pK . Ap), Ap was taken as (L`x c) for 2x2 unit cells
structure.
rcrpK 2sin 22
1 (6.1)
where n
n Ikn 2cs
22
cr LAK (6.2)
where As is the cross-sectional area of the strut, I is the second area moment of inertia
of the strut, and k is a constant related to node type (equals 1 for pin-jointed connections
in bending [51] and 2 for built-in constraints in shear [69]).
Experimentally measured shear FpK of AA5083 and AA3003 micro-trusses
during the load stage (shown in Figures 6.2a and 6.3a) and the corresponding
analytically predicted values are presented in Tables 6.2 and 6.3 along with the
geometrical parameters used in the calculation. Values of K and n are at 25 °C and
strain rate of 10-2 s-1 (Table 5.3). It is worth addressing that in these tables the difference
between these strain rates is negligible (it was also shown in Chapter 4 that the effect of
strain rate sensitivity is insignificant at room temperature). Ap was determined to be
161
1770 and 1855 mm2 for the experimental AA5083 and AA3003 structures, respectively
(L' and c used in the calculation are presented in Table 3.2).
Table 6.2: Analytical prediction and experimental averaged values of FpK
for 2x2 unit cells AA5083 structure with r = 0.019, As =2.436, and I=0.287 sheared at 25 °C and strain rate of 8x10-3 s-1. K and n values at strain rate of 10-2 s-1 was used in the calculation. Geometrical values are from Table 5.2.
Structure K n pK
(MPa)
FpK
(N)
Analytical 675 0.30 0.681 1205
Experimental -- -- 0.87 1552
Table 6.3: Analytical prediction and experimental averaged values of FpK
for 2x2 unit cells AA3003 structure with r = 0.019, As =2.567, andI=0.335 sheared at 25 °C and strain rate of 8x10-3 s-1. K and n values at strain rate of 10-2 s-1 was used in the calculation. Geometrical values are from Table 5.2.
Structure K n pK
(MPa)
FpK
(N)
Analytical 217 0.25 0.295 547
Experimental - - 0.314 583
A comparison between the experimental and analytically predicted FpK in these
Tables indicates a difference, with the experimental values being 28.8% and 6.6%
higher than the analytically predicted ones for AA5083 and AA3003 structures,
respectively. This is might be due to the rotation of trusses in compression, where
rotating trusses may carry more load than non rotating ones. The higher difference in
the case of AA5083 structure than that in the case of AA3003 structure is likely
162
attributed to the effect of cr on the strength of structures failing by inelastic buckling
through the dependency of n on the strain rate (it was shown in section 4.4 that at room
temperature n of AA5083 is sensitive to strain rate, whereas it is insensitive for
AA3003).
The comparison between the analytically predicted shear FpK and those from
simulation load/reload results (from Figure 6.4) structures is presented in Table 6.4 for
both AA5083 and AA3003. In addition, the corresponding geometrical parameters used
in the calculation are also presented in this table. These results were obtained for
structures composed of 2x2 pyramidal unit cells. Values of K and n (which were used in
the input files for the simulation) were obtained from Table 6.1. The predicted FpK
values were then multiplied by 2 to obtain the total force applied on the whole structure.
For these calculations, Ap was determined to be 1398 mm2 and used in both simulation
and analytical calculations for both structures. Ap was calculated using L’ and c given in
Table 3.6.
Table 6.4: Analytical prediction and simulation FpK for 2x2 unit cells of pyramidal structure with r =0.019, As=1.9, and I=0.28 sheared at 25 °Cand strain rate of 8x10-3 s-1. Geometrical values are from Table 5.5.
Structure K n pK
(MPa)
FpK
(N)
EDM AA5083 structures
Analytical 570 0.23 1.923 2688
Simulation - - 1.66 2320
EDM AA3003 structures
Analytical 220 0.25 0.672 939
Simulation - - 0.58 811
163
A comparison of values in these tables reveals that the analytical FpK values are
15.9% and 15.8% higher than those obtained from the simulation for AA5083 and
AA3003 structures, respectively. However, the differences are very close, suggesting
that the small difference might be due to model c, or Hc.
For assessment of strength in 4- cr was analytically predicted for
AA5083 structure only. The prediction was made for a structure of 3x3 unit cells at ts:tf
ratio of 2, using values in Table 3.5. This ratio was selected for the reason that the thin
face sheet at this ratio has insignificant effect on the deformation of the core, as was
shown in section 6.4.2. The calculated results were then compared to cr calculated
using simulation peak loads taken from Figure 6.5 (composed of 3x3 unit cells loaded
in bending at 25 ºC).
In order to predict the contribution of core shear to the strength degradation of
the structure in bending, and also to assess the consistency of analytical and FE
simulation results for the AA5083 structure deformed in bending, further calculations
were made. First, simulation cr was calculated by equation (6.3) using FpK predicted by
FE model:
s
pK
cr AN
F
(6.3)
where N represents the number of load bearing buckled struts. It can be seen in Figure
6.5, that the only struts buckled during bending are those at the ends of the structure (6
struts at each buckled edge). Thus, N in equation (6.3) equals 12. Second, to calculate
the analytical cr flex, has to be calculated first
using equation (6.4):
164
4B.d2
fflex
M(6.4)
flex is the flexural strength (N/mm2) and (B.d2/4=Zp) is the plastic section
modulus (mm3). It is worth noting that Zp was applied to the fully plastic condition due
to the formation of a hinge (Ashby, 2011). Mf is the failure moment of the sandwich
(N.mm), which was calculated using equation (6.5):
4
12LpKF
fM (6.5)
where (L-12) is the span in 4-point bending. FpK is the value calculated from equations
(6.1) and (6.2) [63] using values of K and n of AA5083 at 25 ºC and a strain rate of
2x10-4 s-1 (Figure 4.12a). In the calculation of FpK (FpK= pK . Ap), Ap was taken as (L`xc)
for a 3x3 unit cell structure. The geometrical description is shown in Figure 6.6.
Figure 6.6: A schematic of AA5083 3x3 unit cells micro-truss sandwich. tf is the face sheet thickness, c is the core thickness, Hc (=16.5 mm) is the distance from the middle of the upper and lower face sheets, d is the
sandwich thickness, L (=72 mm) and B (=72 mm) are the span length and width with square cross-section, respectively.
165
The calculated flex from equation (6.4) was then substituted in equation (6.6) [2]
pK cr using equation (6.1) at different
relative densities and ts:tf ratio of 2. For better illustration of the calculation of cr,
Figure 6.7 represents a block diagram for the calculation algorithm:
yflex ff1dL4
BB 2
c3
4 (6.6)
where B3 and B4 are constants belonging to the configuration loads in bending and equal
to 4 and 2 respectively [2]. B, d, and L are sandwich width, thickness and length (mm)
respectively (Figure 6.6). B and L have the same value for a square panel section. f is
the relative volume occupied by face sheet thickness, and corresponds to the ratio of
doubled face sheet thickness to panel height (2tf/d). pK y are the core shear
strength of the structure and yield strength of the material respectively. The latter was
selected to be 113.55 for AA5083 at 25 oC and strain rate of 2x10-4 s-1 (Figure 4.6).
Figure 6.7: Block diagram of calculation algorithm for cr in bending.
Equation (6.5)FpK
Equation (6.4)
Mf
flex
Equation (6.6)
c
Equation (6.1)
cr
166
0
5
10
15
20
25
30
0 0.0005 0.001 0.0015 0.002 0.0025 0.003
Relative density
Crit
ical
buc
klin
g st
ress
, MPa
Theoritical prediction
Simulation
cr are compared in Figure 6.8. Clearly,
both values are in good agreement, confirming that the structure is predicted to fail by
core buckling due to the buckling of 12 struts only.
Figure 6.8: Comparison between simulation and analytical cr for 4-point bending of 3x3 unit cells micro-truss at 25 °C. Prediction shown is for K= 600 and n= 0.3 (Figures 4.12a and 4.13) using equation (6.2). Geometrical
values are from Appendix V.
6.5.2 Comparison of Damage with Compression
The comparison of deformation due to shear and compression was carried out in
term of normalized load carrying capacity, i.e. F/FpK. Normalized experimental and
simulation shear and compressive load-displacement profiles of AA5083 and AA3003
structures are shown in Figure 6.9. The structures consist of four unit cells tested at 25
°C and strain rates of 10-2 s-1 and 8x10-3 s-1 for compression and shear tests respectively.
The experimental compressive and shear load-displacement curves for AA5083 and
AA3003 structures were adopted from Figures 5.4, 5.7 and 6.2a. The simulated shear
load-displacement curves for AA5083 and AA3003 structures were taken from Figure
6.4. Simulated compressive load-displacement curves for AA5083 and AA3003
structures were taken from Appendix V.
(a)
(b)
Figure 6.9: Comparison of normalized load (F/FpK) of 2x2 unit cells pyramidal micro-truss as a function of plastic displacement in compressive
and shear loading modes at 25 °C; (a) experimental, and (b) simulation. Images are for strut buckling in shear and compression loadings.
168
The experimental normalized peak shear strength (F/FpK) and normalized peak
compressive strength of the AA5083 structure are in excellent agreement (Figure 6.9a).
The shear load carrying capacity in the softening region of the AA5083 structure is
higher than that for the structure in compression (Figure 6.9a). This indicates that the
shear strength of these structures decreases more slowly than their compressive
strength. This is attributed to the fact that in shear half of the struts are buckled, while
the other half are stretched, whereas all struts are subject to buckling when these
structures were compressed (Figure 6.9a).
Normalized simulation results are shown in Figure 6.9b, where AA5083 and
AA3003 structures were strained at 11% and 7% in shear respectively, whereas
compressive strain was 17% for both structures. Normalized shear load fell slightly
faster than normalized compressive load. Shear strength of AA5083 and AA3003
displayed a nearly identical normalized load-displacement profile, whereas a small
difference is seen on the compressive profiles. The simulation results suggest that
normalized shear and compressive strength response is insensitive to alloy grade. This
implies that the shear and compression simulation results are applicable to any other
micro-truss structure of similar design as those in this work and made of any aluminum
alloy. A comparison of images from experimental and simulation results shown in
Figure 6.9 indicate that FE model described the core buckling due to shear reasonably
well.
The comparison of normalized load in Figure 6.9 demonstrates the role of core
shear in the deformation behavior of the structures. However, since half of the struts are
carrying load in shear, it is necessary then to estimate the load required to yield half the
struts (FE) using equation (6.1) while the FpK was taken from simulation shear results
(Figure 6.4a). The estimated FE along with the geometrical parameters of the 2x2 unit
169
0
500
1000
1500
2000
2500
0 0.5 1 1.5 2 2.5 3Displacement, mm
Shea
r loa
d, N
Simulation loadEstimated load
cells AA5083 micro-truss structure used in the simulated shear test are presented in
flex of the structure in bending, as
will be shown next.
Table 6.5: Parameters used in calculating FE for AA5083 structure.
cr was calculated using equation (6.3), assuming that N equals 8 (half the
cr was then substituted in equation (6.1) to determine FE.
The comparison is presented in Figure 6.10. It can be realized that the estimated load
represents ~60% of FpK predicted by simulation.
Figure 6.10: A comparison of FpK (simulation) and FE (estimated required to yield half of the struts) for 2x2 AA5083 structure sheared at strain rate of 8x10-3 s-1 and 25 °C. Load-displacement curve was taken from Figure 6.4a.
Structure r FpK
(N)
As
(mm2)
cr pK
(MPa)
FE
(N)
AA5083 0.019 2320 1.9 152.6 1.04 1449
170
The struts in tension will rapidly lose load bearing capacity after occurrence of
neck axial, was calculated,
enabling examining the criterion of diffuse necking that may occur when axial = n [100].
Figure 6.11 shows a sketch of the core in shear, where struts in tension and compression
are represented. The calculation was made using equation (6.7) at n=0.23 taken from
Table 6.4.
Figure 6.11: Sketch shows the deformation profile of struts in shear.
axial (6.7)
e shear displacement, which is 3 mm in the simulated shear
this angle was 45oc) (equals 10.3o
axial was found to
be 0.09. To estim buckling, the tangent
modulus Et,, (the slope of the stress-strain curve, 1-Kdd nn ) was used assuming K =
570 and n = 0.23 (Table 6.4). Et, cr = 152.6 (Table 6.5) was obtained from the
following equation [28, 55, 69, 145]:
2cs
t22
cr LA. IEk
(6.8)
Strut in compression
Strut in tensile
171
Where As, I, and Lc are the cross-sectional area, second area moment of inertia (Table
6.4) and length member of the strut respectively, k value was set at 2 for shear tests
with fixed end condition [55, buckling was found to be 0.071.
It can be concluded that strains in struts undergoing tensile and compressive
loading in shear are close, suggesting that levels of deformation will also be similar. In
addition, the value of axial (0.09) is lower than that of n (0.23), suggesting that no
diffuse necking in the struts is possible at these conditions.
6.5.3 Comparison with Foams
This section presents comparisons of the current experimental results of the
degradation behavior of stiffness and strength for EDM AA5083 and brazed AA3003
structures in shear with results from previous work, and with simulation results. The
performance was then compared to that of foam for stiffness and strength assessment
purposes. Shear data used for comparison were obtained at 25 °C. The comparison was
performed through normalized shear stiffness and strength, i.e. Gc/Es and pK y
respectively.
The normalized shear stiffness is defined as the ratio of core shear stiffness
modulus (Gc) to Young’s modulus of the material (Es). Gc was determined from the
ratio of yield strength to strain at yield (obtained from equation 5.4), where the yield
strength is defined as the shear load at yield divided by the area (L’x Hc). Es values are
adopted from Table 5.4 for both EDM AA5083 and brazed AA3003 structures. pK y is
defined as (the ratio of pK was
calculated from Figures 6.2 and 6.3 and y was adopted from Table 5.3 for both EDM
AA5083 and brazed AA3003 structures. Data for PCM and foam in Figures 6.12 and
6.13 were obtained using models in the literature [31, 106].
172
The behavior of the experimental Gc/Es, and pK y of the current micro-truss
structures with respect to foam are presented in Figure 6.12. It can be seen in Figure
6.12a that despite the sharp decline in Gc/Es of the AA5083 structure with increasing
strain, the structure retained stiffness greater than that of a foam of equivalent r , and
only reached that of a foam when it was sheared at 11%. Gc/Es of the AA3003 structure
also exhibited a sharp decline, but never dropped below that of the foam even at a strain
of 9%. These results indicate that both structures have better shear stiffness than a foam
of equivalent r .
Both micro-truss structures pK y behavior with increasing
pK y values well above those of a foam at equivalent r at all
pK y of AA5083 and AA3003 structures is higher than that of a foam by
~50% at the highest strains (Figure 6.12b). It can be concluded that despite the fact that
both micro-truss structures experienced a drastic decrease in their mechanical properties
with increasing shearing strain, the micro-truss structure outperformed a foam structure
at equivalent r . However, it can be realized from Figure 6.12 that r has very limited
effect on the Gc/Es, and pK y of both structures.
(a)
6%
7%9% 11%
0.00001
0.0001
0.001
0.01
0.01 0.1Relative density
Norm
aliz
ed s
tiffn
ess
PCM [31]Foam [106]AA5083-present workAA3003-present work
173
(b)
Figure 6.12: Comparison between the experimental normalized mechanical properties of aluminum pyramidal unit cell and commercially available
open cell aluminum foams at equivalent r at 25 °C and strain rate of 8x10-3
s-1; (a) Gc/Es, and (b) pK y.
To illustrate the shear performance of the current structures compared with those
published in the literature, a comparison between the present results shown in Figure
6.12 and those reported by Queheillalt et al. [63] is given in Tables 6.6 and 6.7. The
performance of the present structures appears inferior to that of AA6061 structure. This
is attributed to different material properties and design, as has been discussed earlier in
section 5.5.2. It is worth addressing that the degradation in stiffness (Gc/Es) for the
partially damaged AA3003 and AA5083 structures shown in Figure 6.12a and Table 6.7
is relatively higher than that of AA6061.
9%
6%
0.00001
0.0001
0.001
0.01
0.1
0.01 0.1Relative density
Norm
aliz
ed s
treng
th
PCM [31]Foam [106]AA5083-present workAA3003-present work
11%7%
174
Table 6.6: Comparison of strength parameters, geometrical and material properties of current experimental work of AA5083 and AA3003 with a single pyramidal core with that of AA6061 sheared at 25 °C [63].
Table 6.7: Normalized shear stiffness and strength of partially damaged single pyramidal AA6061 core at 25 °C [63].
Strain
(%)
Gc
(MPa)
Gc/Es pK
(MPa)
pK y
7 390 0.0057 5.95 0.02219
9 355 0.0051 5.94 0.02218
11 323 0.0047 5.894 0.02199
Geometrical parameters used in shear simulations are given in Table 6.4, and the
simulation results of Gc/Es, and pK y of the structures are shown in Figure 6.13.
Mechanical properties used in the simulation were obtained from Table 6.4. It should be
noted here that Es values were not the real ones but that using them in the models does
not change the basic behavior which is the main focus. It can be seen that Gc/Es
Properties AA6061 AA5083 AA3003
r 0.062 0.019 0.019
c (mm) 19.1 15.0 15.4
Lc (mm) 24.6 25.7 26.3
ts (mm) 3.2 1.46 1.53
(Deg) 50.77 35.7 35.98
L (mm) 22 29.5 30.1
Ap (mm2) 420 442.5 463.8
Es (GPa) 69 70 70
y (MPa) 268 103 45
(Gc/Es) initial loading 0.007 0.0019 0.0016
pK y) initial loading 0.022 0.0085 0.007
175
obtained from simulation (Figure 6.13a) and pK y (Figure 6.13b) of AA5083 are in
good agreement with the experimental values shown in Figure 6.12. It is useful to
address that the simulation results of AA5083 and AA3003 structures (Figure 6.13)
were not included in Figure 6.12 due to different design features (different inclination
angles).
The simulation of the AA3003 structure predicted initial Gc/Es to be 32% greater
than was measured in the experiment. This can be rationalized by the effect of
annealing on the stiffness of the structure, which was not accounted for in the
simulation. The effect of annealing in this work agrees well with previous results [148]
where 20-30% degradation was observed on the strength of annealed AA3003 micro-
truss structures compared to non-annealed ones. The simulation results indicate that
Gc/Es and pK y of both structures responded in a similar fashion to increasing strain.
Furthermore, the results suggest that r has a minor effect on Gc/Es and pK y.
(a)
0.00001
0.0001
0.001
0.01
0.01 0.1Relative density
Norm
aliz
ed s
tiffn
ess
PCM [31]Foam [106]
AA5083-simulationAA3003-simulation
6% 7%9%
11%
176
(b)
Figure 6.13: Comparison between the simulation normalized mechanical properties of aluminum pyramidal unit cell and commercially available
open cell aluminum foams at equivalent r of 0.019, 25 °C and strain rate of 8x10-3 s-1; (a) Gc/Es and (b) pK y.
According to the simulation results, although the degradation in Gc/Es pK y
with increasing strain is rapid, the structures retained a superior performance compared
to the foam at all strains. However, it can be realized that Gc/Es degrades more quickly
in experiment. In general, it can be noted that the FE model predicted the degradation
pattern of Gc/Es and pK y of the core with reasonable accuracy.
6.6 Conclusions
This Chapter presented experimental and simulation results of AA5083 and
AA3003 micro-truss structures tested in load and reload shear and bending. The reload
stage was carried out independently of the load stage, using the same pre-strained
structures (partially damaged in the load stage). The results of this chapter provided
clear insights into the effect of the pre-straining level on the degradation of stiffness and
Good potential for heat exchange process and also in fluid flow passages. These materials are characterised by their capabilities of absorbing large amounts of mechanical energy
[43]
High strength aluminum alloy
Honeycomb Pin joints offering no rotational resistance from member to member or to the face
Multifunctional application of heat transfer and mechanical loads or compression and tension
[27]
Not mentioned Two-dimensional honeycombs:- Hexagonal- Square- Triangle
Enhance impact/blast energy absorption, noise attenuation, catalytic activity, filtration efficiency, electrical energy storage, or act as the host for thein-growth of biological tissue
[20]
204
Materials Core shape Joints type Applications ReferencesBeryllium-copper casting alloy (Cu-2%Be). These alloys are characterized as:
High thermal conductivityHigh ductile (failure in aluminum alloys under certain conditions was attributed to the relatively low ductility of cast aluminum alloys.Good strain properties
Tetrahedral Prototype-Casting method
Good for high tension and compression multifunctional applications. Suitable for heat transfer process
[21]
Not mentioned Not mentioned Many bonding processes are available to fabricate different structures. For some materials, resistance welding can be used. For titanium alloys diffusion bonding has been successful. Brazing methods can be used for aluminum alloys. For many stainless steels, superalloys and copper alloys a transient liquid phase (TLP) process can be used.
Circular Tubes Seam welding (CS-1020). Aluminum alloys were seamless.
Effective for energy absorption applications
[60]
Not mentioned 3D Kagomé Pin-jointed Actuation applications [8]-Metals (Al,Mg,Ti)-Polymers-Elastomers-Glasses-Ceramics
Different shapes depending on the application.
Not mentioned For aircraft, automobile and sport equipments applications (multifunctional purposes)
[34]
AA6061 sheets (for its excellent brazing characteristics, high yield strength-to-weight ratio when precipitation hardened, high thermal conductivity and long history of successful applications).
Tetrahedral Air brazing method Multifunctional applications in cross heat exchange, shape morphing, and high intensity dynamic load protection
Transient liquid phase (TLP) bonding and brazing method
For heat transfer applications [14]
206
Materials Core shape Joints type Applications ReferencesSS304 Pyramidal Brazing using a mixture
of a polymer-based cement (Nicrobraz Cement 520) and a Ni-22Cr-6Si braze powder (Nicrobraz 31)
Heat transfer and energy absorption (blast resistance)
[51]
Not mentioned 2D triangular core, and 3D tetrahedral and honeycomb cores
Not mentioned Applications with a combination of transverse shear, bending and crushing stress
[22]
SS304 Square honeycomb Brazing method For energy absorption applications [19]
AA6061 sheets - Tetrahedral- Plain square weave
Brazing method For heat transfer applications, such as in heat exchangers giving high convection exchange with low pressure drop.
[30]
SS304 - Diamond- Square honeycomb
Brazing method using Nicrobraz 51 alloy
In cross flow heat exchange process
[55]
SS304 Pyramidal Vacuum brazing method For aerospace applications as a replacement (candidate) of the solid face sheets sandwich panels of lightweight structure
[46]
Not mentioned Clamped sandwich beams
Not mentioned Air and underwater shock wave resistance
[10]
All-metal sandwich plates
Square honeycomb Not mentioned Core crushing strength and energy absorption under uniform impulsive pressure load applications
[17]
SS304 - Prismatic- Diamond
Brazing using an alloy of Ni-Cr 25-P10 (wt.%)
Energy absorption and shock resistance sandwich construction
[23]
207
Materials Core shape Joints type Applications ReferencesNot mentioned Sandwich beams Not mentioned For energy absorption applications [37]AA3003 for cores (Al-1.2%Mn-0.12%Cu) and aluminum 6951 alloy for face
Pyramidal Furnace brazing technique
Multifunctional applications [69]
SS304 sheets and cores of stainless steel AL6XN
Pyramidal Brazing Marine applications [83]
SS304 Pyramidal (multilayer structure)
Transient liquid phase (TLP)
Underwater blast loading [45]
- SS- Copper
Woven textile core with diamond or square pores
- Transient liquid phase (TLP) for stainless steel cores.
- Brazing for copper cores.
Multifunctional purpose of load bearing and heat transfer applications
[16]
SS304 Pyramidal (multilayer structure)
Transient liquid phase (TLP)
Underwater blast loading [50]
SS304 with polymer and ceramic fillers
Pyramidal (multilayer structure)
Brazing method For ballistic response to moderate velocity impact by a spherical projectile
[6]
AA3003-H14 (This alloy is widely used in industry because of its low cost, ability to work harden, and good formability).
Pyramidal A single core unit was used, no welding was introduced.
Multifunctional applications such as cross flow heat exchange, shape morphing and high intensity dynamic load protection. They are also promising candidates for impact energy absorption applications
[68]
AA3003 Pyramidal Brazing method Multifunctional applications of high strength but lightweight structures
[28]
AA6061 cores placed between AA6951 face sheets with aluminum-silicon 4343 braze alloy
Tetrahedral Furnace brazing technique in air at 595±5 oC
Multifunctional applications, such as cross flow heat exchange, shape morphing and high intensity dynamic load protection
[62]
208
Materials Core shape Joints type Applications ReferencesAA6061 Pyramidal lattice
sandwich structuresA product of extrusion and electro discharge machining
Suitable for weight –sensitive applications
[63]
AA6061 coated with an electroless nickel layer
Truncated-square honeycomb
Brazing method For heat transfer applications [39]
AA3003-H14 Pyramidal Brazing Multifunctional for weight-limited engineering applications, such as panel stiffening in sandwiches
[28]
SS304 - Pyramidal (multilayer structure)
- Honeycomb (square and triangular)
- Triangular and diamond corrugated cores (multilayer structure)
- Prismatic (multilayer structure)
Vacuum brazing using brazing alloy powder Nicrobraz 51 alloy for 1 h at 1050 oC
and
Transient liquid phase (TLP) using a brazing paste (Wall Colmonoy Nicrobraz 51 alloy) at 1050 oC
Straight strut only, no specified core design was used
No joints Good potential for heat exchange process and also in fluid flow passages. These materials are characterized by their capabilities of absorbing large amounts of mechanical energy
[80]
209
Materials Core shape Joints type Applications ReferencesSS304 - Pyramidal
(multilayer structure)
- Honeycomb (square and triangular)
- Triangular and diamond corrugated cores (multilayer structure)
Vacuum brazing using brazing alloy powder Nicrobraz 51 alloy for 1 h at 1050 oC
For heat transfer applications [33]
SS316 L Octahedral and pillar-octahedral
Selective laser melting technique
Not mentioned [64]
AA2A12-T4 Pyramidal Film adhesive (J-272) Enhance impact/blast energy absorption, noise attenuation, catalytic activity, filtration efficiency, electrical energy storage, or act as the host for thein-growth of biological tissue
[65]
210
APPENDIX II
211
Brazing of Aluminum Alloys
Brazing is a good choice for bonding of aluminum alloys, which are difficult to
join by the traditional welding process. Brazing is a state of formation solidified joint
between metallic materials. It is complex due to the presence of an oxide layer on the
cladding preventing free flow of the melted metal into the joint, hindering the
wettability of the metal surfaces. Aluminum brazing is used for joining aluminum alloy
parts by applying filler alloys which have lower melting points than the parent
from brazing of AA3003-AA4343 alloys showed that joint formation is controlled by
surface tension, with less influence of gravity, dissolution phenomena and subsequent
solidification. Currently, brazing technology gained a wide use in truss constructions for
being reliable and efficient [15,30,32,38]. Many interrelated factors have to be taken
into account when designing a joint in a truss structure that is to be manufactured by
brazing. The five most important considerations are:
1. The type of parent metals to be joined
2. The position of these parent materials relative to each other in the joint
3. The type of filler material to be used to make the joint
4. The fixing of the components
The majority of brazing is carried out in air. To be a successful, a chemically
clean surface is provided at the faying surfaces of the joint at brazing temperature so
that the filler material will wet and flow into and through it. It was addressed [29] that
the presence of microscratches on the surface of materials to be wet by molten brazing
alloy is not a bad thing; they provide pathways that can enhance the flow of the molten
brazing material. Therefore, it is suggested to arrange the direction in which the
212
scratches are parallel to the desired direction of alloy flow in order to assist in the flow
of alloy.
A considerable number of applications still utilise flame brazing. Flame brazing
is characterized by the fact that the rate of joint production and the quality of the
finished joint are directly under the control of the operator. Accordingly, the rate of
production and the appearance of the finished joints will be constantly varying.
However, one of its main attractions is being a very flexible method. On the other hand,
there is no doubt that today the largest numbers of brazed joints are made in protective-
atmosphere brazing furnaces, which is nominated as furnace brazing. This method
could be divided into two major techniques according to the pressure inside the furnace:
vacuum or atmospheric brazing [29].
With the rapid development of welding technology, a new brazing method was
introduced recently using a CO2 laser beam as an energy source. The laser energy can
penetrate into the aluminum which yields rapid local surface melting on the parts to be
joined. This method is nominated as Laser brazing [76]. Infrared brazing is another
method that has been originally developed at the University of Cincinnati for high
temperature materials [77]. In this method, infrared energy generated by heating a
tungsten filament in a quartz tube as the heating source is commonly applied. In
general, materials welded using infrared brazing have higher melting temperature than
those welded by CO2 laser brazing. In furnace brazing method, the heat is transmitted to
the joints by radiation from the heating elements or from the walls of a gas-tight muffle
that has been heated externally. Regardless of the employed heating method, the
development of the correct heat pattern is one of the fundamental requirements for
producing a satisfactorily brazed joint. The development of the required temperature
gradient across the joint requires that controlled heating of the whole joint is
213
undertaken. The objective is to ensure that all parts of it attain a temperature that is at
least equal to the working temperature of the chosen filler material. Unlike flame
heating where temperature control of the parts can be quite troublesome to achieve, with
furnace brazing it is very easy to ensure that overheating of an assembly cannot occur.
This is because the temperature control of the furnace can be set to a precise value.
The rate of heating of the joint depends on a number of factors. Some of the
more important ones are the masses of the components, the intensity of the heat source
being used, and the thermal conductivity of the materials that compose the joint [29].
Managing these factors in the correct manner produces satisfactory joints as shown in
Figure A2-1 for line-contact type. While they can be fed by means of brazing filler
material preplaced in the angle, they offer an ideal application for clad sheet.
Figure A2-1: A satisfactory brazed joint type for aluminum alloy [29].
Both flame brazing and furnace brazing can be used for brazing of aluminum
alloys. However, flame brazing of aluminum alloys have potential metallurgical
difficulties relating to the composition of the filler materials and parent metals which
demands particular attention, especially when the alloys contain magnesium. This is
because the magnesium-containing alloys have solidus temperatures of about 616 oC. If
such materials are overheated, they are prone to incipient grain-boundary melting,
214
commonly known as the orange-peel effect [29]. On the other hand, controlled
atmospheric furnace brazing of aluminum alloys has evolved as the leading technology
for manufacturing of aluminum parts for automotive industry. Its advantages can be
summarized as follows:
1. There is a successful removal of the tenacious layer of aluminum oxide that is found
on the surface of the parent material.
2. The process works at atmospheric pressure.
a. There is no reaction between the flux and the aluminum alloy substrate.
b. The flux residue has almost zero solubility in water and so does not hydrolyze.
3. A noncorrosive flux is employed
4. There is no need to undertake any post-braze treatment of the assembly.
5. There are no flux-related corrosion issues.
Vacuum brazing of aluminum alloys is not as widely practiced as the
atmospheric brazing. It is used mostly with aluminum alloys containing magnesium of
1-2%. High vacuum is required for this method, and hence having a very sophisticated
pumping system is critical which must be 100% leak free. For aluminum alloys
containing <2% Cu, brazing is not a problem. Any convenient economic brazing
method is applicable. Table A2-1 lists the brazeability of the most common aluminum
alloys showing their behavior toward brazing.
215
Table A2-1: The Brazeability of the various families if aluminum-base material [29].
Parent Material Brazeability CommentsAA1000 Series
AA2000 Series
AA3000 Series
AA5000 Series
AA6000 Series
AA7000 Series
Cast materials
Good
Not recommended
Good
Limited
Good
Not recommended
Caution: Can be very difficult
No real problems.
Brazing results is an irreversible metallurgical
deterioration in the parent material.
No real problems.
The difficulties of brazing increase as the Mg
content rises above 0.7%.
Caution: There is a loss of tensile strength;
always check the solidus temperature of the
parent material; post-braze ageing is a
possibility.
Brazing results is an irreversible metallurgical
deterioration in the parent material.
Brazing with BS EN1044 Type AL104 is impossible; it would be worth trying BS EN1044 Type AL201, but even this will probably be unsatisfactory
In most cases, brazing filler metals do not have a single melting point but melt
over a specific temperature range, as shown in Table A2-2. The temperature at which a
brazing alloy can be used to make a joint must always be higher than the temperature at
which it begins to melt.
216
Table A2-2: Properties of some aluminum alloys [29].
Material code
Solidus (°C) Liquidus (°C) Comments
AA1070
AA1145
AA3003
AA3005
AA3102
AA3105
AA6061
AA6063
AA6951
640
640
643
640
645
635
616
616
616
655
655
654
655
655
655
652
652
654
Brazing poses no real problems.
Brazing poses no real problems.
Brazing poses no real problems.
Caution: This material can contain
up to 0.6% Mg.
Brazing poses no real problems.
Caution: This material can contain
up to 0.8% Mg and might be difficult
to wet.
Caution: This material contains
between 0.8 and 1.2% Mg and will
be difficult, but not impossible, to
wet.
Caution: This material can contain up to 0.9% Mg and might be difficult
to wet.
Caution: This material can contain up to 0.8% Mg and might be difficult to wet.
In this Table, the solidus temperature of an alloy is the temperature at which it
begins to melt when being heated from room temperature. On the other hand, the
liquidus temperature of an alloy is the temperature at which it becomes completely
molten. The temperature difference between the solidus and liquidus temperatures of an
alloy is known as its melting range or plastic range. In those rare situations where the
solidus and liquidus temperatures coincide and where, in consequence, there is a
melting range of 0 °C, the material is known as a eutectic.
Once a brazing filler material is heated to its solidus temperature, it begins to
melt. As the temperature is gradually increased, more of the alloy becomes molten until,
217
at its liquidus temperature, the material becomes 100% liquid. Throughout the melting
range of the alloy, the ratio of the liquid phase to the solid phase increases as the
temperature rises; the fluidity of the alloy also increases. This concept is illustrated in
Figure A2-2. At a temperature above the solidus of the filler material, the molten filler
material can possess a level of fluidity sufficient to enable it to flow into a capillary gap
and make a joint. The temperature at which this occurs is known as the working
temperature of that filler metal [29].
Figure A2-2: Representation of working temperature.
The selection of brazing filler metal primarily relies on its wettability.
Aluminum alloys are often welded with filler metals that do not match the parent metal
in some or all of properties, i.e. composition, mechanical properties and appearance
[29]. Among the factors that affect wettability of filler metal, the compositions of
brazing filler metal and base metal are both decisive. If the filler metal and base metal
can dissolve mutually or form an intermetallic compound, the melted filler metal can
wet the base metal better; if the melted filler metal contains surfactant materials that can
decrease surface tension markedly, the wettability of filler metal can be improved.
Wettability has been the subject to an intensive study carried out by Wang and co-
workers [78] who studied the influence of rare earth elements on the microstructure and
Working temperatureMelting range
Solidus100% Solid
100% Liquid
Liquidus
218
mechanical properties of AA6061 alloy welded using vacuum brazing method. It was
found that rare earth elements improve the wettability of welding process through
increasing the strength of vacuum brazed joints [79].
A molten brazing filler material that possesses an appropriate level of fluidity
will always flow toward the hottest part of a capillary joint even if this means that the
direction of flow is against the force of gravity. It follows that to produce a joint, a
molten filler material that has moderate-to-good flow properties must be drawn by a
combination of capillary attraction and temperature gradient into and through the joint.
For this to occur, the mating surfaces of the joint have to be parallel, relatively close
together, as shown in Figure A2-3, and chemically clean [29]. When brazing is
undertaken in air, the production of a chemically clean surface is normally provided by
the use of fusible flux.
Figure A2-3: Capillary attraction (Pc) as a function of gap width [29].
The corollary of this Figure (A2-3) is that, for effective flow and filling of the
joint to result, the joint gap at brazing temperature should lie in the range 0.05 to 0.2
219
mm. This range is acceptable for any brazing process where a fusible flux has to be
employed [29].
220
APPENDIX III
221
Melting temperature determined by Differential Thermal Analysis (DTA)
AA5083
AA3003
Figure 1: Melting temperatures by DTA.
222
Calculations of Geometrical Terms
1. Truss Dimensionsc: Measured experimentally
: Measured experimentally
Lc: Measured experimentally
As: 2 . Lc . , and, c= Lc sin
So, tangent ='
2Lc , and
'2L
c
The results of experimental calculations of c for the unit cell of pyramidal
micro-truss sandwich were presented in Table 3.2 in chapter 3.
2. Strut DimensionsTable 3.2 presents the average values of strut dimensions (a, b, ts), which were used for
further calculation of geometrical terms, e.g. As and I. In the case of experiment, these
dimensions were measured using a digital caliper, whereas in simulation, they were
obtained from equation (3.3) in chapter 3 with a relative density of 0.019:
sincosLA222
c
sr (3.3)
Taking into account that (Lc and ) were considered to be constants in simulation
calculations. However, it is very important to mention that equations (As=21 a.b) and
(48
ba 3
I ) were then used to calculate the cross – section area and second moment of
area of the strut. The results were presented in Tables 5.1, 5.2, and 5.5 for the
experimental and simulation sections in chapter 5.
223
APPENDIX IV
224
True stress-strain plots used for n and K calculations – AA5083
(a) (b)
(c) (d)
Figure 1: True stress-strain profiles of AA5083 and different strain rates; (a) 2x10-4 s-1, (b) 10-3 s-1, (c) 10-2 s-1, and (d) 10-1 s-1.
225
True stress-strain plots used for n and K calculations–AA3003-1 side
clad
(a) (b)
(c) (d)
Figure 2: True stress-strain profiles of AA3003-1 side clad at different strain rates; (a) 2x10-4s-1, (b) 10-3 s-1, (c) 10-2 s-1, and (d) 10-1 s-1.
226
True stress-strain plots used for n and K calculations–AA3003-2 side
clad
(a) (b)
(c) (d)
Figure 3: True stress-strain profiles of AA3003-2 side clad at 25 °C and different strain rates; (a) 2x10-4s-1, (b) 10-3 s-1, (c) 10-2 s-1, and (d) 10-1 s-1.
227
True stress-strain plots at different temperatures used for n and K
calculations
(a)
(b)
Figure 4: True stress-strain profiles; (a) AA5083, and (b) AA3003-2 side clad at elevated temperatures.
228
0
50
100
150
200
250
300
350
0 0.05 0.1 0.15 0.2
True strain (mm/mm)
True
str
ess
(MPa
)
specimen 1Specimen 2Specimen 3Fitting
0
20
40
60
80
100
120
140
160
0 0.1 0.2 0.3
True strain (mm/mm)
True
str
ess
(MPa
)
Specimen 1Specimen 2Specimen 3Fitting
Fitting curves to the experimental true strain-stress tensile profileusing values of n and K obtained by Hollomon model
These figures show an example of fitting curves using values of n and K taken from Figures 4.12a and 4.13 in section 4.3.3.
(a)
(b)
Figure 5: Fitting curves of true stress-strain data at 25 °C and strain rate of 2x10-4
s-1; Fitting is with (a) K= 600 and n= 0.31 for AA5083, and (b) K= 206 and n= 0.225for AA3003-2 side clad.
229
Logarithmic plots for m calculation – AA5083
(a)
(b)
Figure 6: Strain rate sensitivity for AA5083 at 25 °C for strain rate rangesof; (a) 2x10-4 -10-1 s-1, and (b) 10-3-10-1 s-1.
230
Logarithmic Plots for m Calculation – AA3003-2 side clad
(a)
(b)
Figure 7: Strain rate sensitivity for AA3003-2 side clad at 25 °C for strain rate ranges of; (a) 2x10-4 -10-1 s-1, and (b) 10-3-10-1 s-1.
231
Logarithmic plots for m calculation – AA3003-1 side clad
Figure 8: Strain rate sensitivity for AA3003-1 side clad at 25 °C for a strain rate range of 10-3 to 10-1 s-1.
232
Table 1: Strain rate sensitivity (m) values taken from the literature for AA5083.
Table 2: Strain rate sensitivity (m) values taken from the literature for AA3003.
Alloy Temper type Strain rate(s-1)
Temperature (ºC)
m Comments References
Cast commercial AA3003
Casting technique N/A 288 0.01 Annealing [132]
AA3003 Annealed between(520-580 oC)
1.2x10-4 &1.2x10-2
200 0.016 non-heat-treatable
[133]
AA3003 Homogenized 30 475 0.106 [134]
AA3003 H111 8x10-3 25 - 260 0.003 –0.08
[109]
Commercial cast billet AA3003
Annealed at 600oC for 3h then at 400 oC then at 530oC for1h
530 N/A m for AA3003 is lower thanthat for AA5083
[130]
234
0
50
100
150
200
250
300
350
0 0.05 0.1 0.15True strain (mm/mm)
True
str
ess
(MPa
)
Specimen 1Specimen 2Specimen 3Fitting
Fitting curves of true strain-stress dataThese figures were used in the theoretical prediction of inelastic buckling in section
5.3.2. As can be seen, the fitting to the experimental data for both AA5083 and AA3003
is very good. As a result, these models were found to be adequate. In addition, these
models have been widely used for describing the work hardening behavior of micro-
truss structures [28, 55, 71] and material characterization at room and elevated
temperatures [109, 120].
Figure 9: Fitting of true strain-stress curves of AA5083 at 25 °C and strain rate of 10-2 s-1. Fitting are for K= 675 and n n).
Figure 10: Fitting of true strain-stress curves of AA5083 at 300 °C and strain rate of 10-2 s-1. Fitting are for K= 76 and n o n).
235
0
20
40
60
80
100
120
140
160
0 0.05 0.1 0.15 0.2 0.25
True strain ( mm/mm)
True
str
ess
(Mpa
)
Specimen 1Specimen 2Specimen 3Fitting
Figure 11: Fitting of true strain-stress curves of AA5083 at 500 °C and strain rate of 10-2 s-1. Fitting are for K= 12.43 and n o + K. n).
Figure 12: Fitting of true strain-stress curves of AA3003 at 25 °C and strain rate of 10-2 s-1. Fitting are for K= 217 and n . n).
236
Figure 13: Fitting of true strain-stress curves of AA3003 at 300 °C and strain rate of 10-2 s-1. Fitting are for K= 30 and n n).
Figure 14: Fitting of true strain-stress curves of AA3003 at 500 °C and strain rate of 10-2 s-1. Fitting are for K= 13.5 and n n)
237
y = 84.694x-0.0697
R2 = 0.2136
0
50
100
150
200
0.0001 0.001 0.01 0.1
Strain rate ( s-1)
Yiel
d sr
engt
h,(M
Pa)
y = -3.4663Ln(x) + 28.777R2 = 1
0
50
100
0.0001 0.001 0.01 0.1
Strain rate ( s-1)
Yiel
d sr
engt
h,(M
Pa)
Interpolated data at a strain rate of 8x10-3 s-1 and 25 °C for AA5083 and AA3003-2side clad used in Chapter 6 for calculating values of tensile parameters ( y, K, and n) presented in Table 6.1.
(a)
(b)
Figure 15: Interpolated va y) at strain rate of 8x10-3 s-1
and 25 °C; (a) AA5083, and (b) AA3003-2 side clad.
238
y = 555.87x-0.0053
R2 = 0.0117
0100200300400500600700800
0.0001 0.001 0.01 0.1
Strain rate ( s-1)
K, (
MPa
)
y = 0.1843x-0.0474
R2 = 0.1842
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.0001 0.001 0.01 0.1
Strain rate ( s-1)
Stra
in h
ardn
ing
expo
nent
, n
(a)
(b)
Figure 16: Interpolated values for AA5083 at strain rate of 8x10-3 s-1 and 25°C; (a) K and (b) n.
239
y = 0.295x0.0316
R2 = 0.973
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.0001 0.001 0.01 0.1
Strain rate ( s-1)
Stra
in h
ardn
ing
expo
nent
, n
y = 246.04x0.0234
R2 = 0.8903
195200205210215220225230235240
0.0001 0.001 0.01 0.1
Strain rate ( s-1)
K, (
MPa
)
(a)
(b)
Figure 17: Interpolated values of AA3003-2side clad at strain rate of 8x10-3
s-1 and 25 °C; (a) K and (b) n.
240
APPENDIX V
241
Simulation compressive load-displacement curves of pyramidal micro-truss structure containing 2x2 unit cells with r of 0.019 at 25 °C and strain rate of 10-2 s-
1, used in section (6.5.2) (corresponding to Figures 5.22 and 5.23).
(a)
(b)
Figure 1: Simulation compressive load-displacement curves of pyramidal micro-truss structure containing 2x2 unit cells with r of 0.019 at 25 °C and
strain rate of 10-2 s-1; (a) AA5083 and (b) AA3003
242
Table 1: Geometrical values of AA5083 structure in 4-point bending at ts:tf=2 used in Chapter 6 for calculating cr (simulation and analytical) and flex presented in